Flight Instructor reported an NMAC after receiving a late hand off by ATC while flying in congested airspace.

Narrative

I am a certified flight instructor (CFI; CFII) and was flying a training lesson with a student pilot (private pilot candidate). Our home airport is ZZZ which is a class D located under a Class C (ZZZ1) and overlapping with an additional class D (ZZZ2). To note; ZZZ1 is a training facility for ATC and ZZZ2 does NOT have radar contact. ZZZ is home to at least 4 flight schools with a significant number of student pilots flying in congested airspaces daily. Additionally; the proximity between these 3 airports is extreme with ZZZ and ZZZ2 being only 6 NM apart and with the departure end of Runway X at ZZZ2 being close to the Arrival corridor for Runway Y at ZZZ. Our Requested route for day and time of this flight was 'VFR to ZZZ2 for pattern work.' The active runway at ZZZ was Runway Y(left traffic) and the active Runway at ZZZ2 was Runway X. Runway X at ZZZ2 is typically a right traffic pattern. My student and I were instructed to make a 'left downwind (to downwind) departure' leaving ZZZ and navigating into ZZZ2 airspace. This is a directive that is often given to aircraft navigating between ZZZ and ZZZ2. Upon our departure my student and I heard a position report from a company aircraft (a student pilot solo) 10 miles out to the south inbound for landing at ZZZ. The aircraft was cleared to land straight in on runway Y at ZZZ in that initial call which would also keep him in close proximity of ZZZ2 and active on the ZZZ frequency for the remaining time. Additionally my student reported to ZZZ ATC that we had ZZZ2 field in sight and we were instructed to contact ZZZ2 Tower. Following our initial call to ZZZ2 Tower my student was instructed to join the left downwind for runway X and report midfield. My student navigated east towards ZZZ2 to join the downwind for runway X near the departure end of runway X with intentions of joining the downwind as advised and then making a position call at midfield. I communicated to my student to navigate 'westbound' and 'to the right' to ensure spacing and safe pattern entry. I noticed my student drifting closer to the departure corridor and near the arrival corridor and as previously noted the airfields are only 6 NM apart with the arrivals and departure corridors being close. Upon seeing the drift and noting closer proximity to this 'shared area' (near ZZZZZ; an outer marker for ILS runway Y at ZZZ). I again instructed my student to navigate 'to the right' for ease of communication and began to take control of the aircraft in an effort to steer west bound. At the time my student initially resisted the control input and I again communicated the need to turn west bound and announced my controls. As I took over full control of the aircraft and started a right turn west bound our left wing lifted. Given the nature of the aerodynamics of a turn in a low wing airplane the high left wing left a blind spot. Simultaneous to all this the ATC Controller at ZZZ2 communicated to us urgently to turn west-bound immediately (a correction I was already initiating. With the urgency in his tone I executed a steeper turn and added power to navigate west. Upon rolling out west bound ATC Tower at ZZZ2 communicated to our aircraft that we were a near miss with jet traffic that was inbound on final for Runway Y at ZZZ. To note; I was never advised by ATC at ZZZ2; NOR by ATC at ZZZ upon leaving their frequency; of any inbound traffic for runway Y at ZZZ. It is standard practice for ZZZ1 Approach to communicate with ZZZ and ZZZ2 of any inbound traffic in anticipation of 'handing them off' to the 3 local Towers. Unfortunately; it is also known that 'ZZZ1 hands traffic off late' and I have often times been on the ZZZ Tower frequency to hear the ATC controllers communicating with inbound pilots about 'aircraft inbound on final that (they're) not talking to 'because ZZZ1 hasn't turned them over' needing to find ways to navigate traffic out of the way. On this evening the time between being handed off from ZZZ; to our initial contact with ZZZ2 Tower; and our being told to turn west bound immediately was a matter of 2-3 minutes. At no time did my student or I hear or acknowledge traffic inbound for runway Y at ZZZ. At no time was I able to look for; see or avoid Jet traffic coming inbound to ZZZ (directly between the overlapping segment of ZZZ3 and ZZZ airspace) as the left wing was high and I was already putting in corrective action to navigate our aircraft west bound. Additionally as SOON as a I was told by ATC at ZZZ2 to navigate west bound I fully executed the directive. While still on with Tower at ZZZ2 I was advised a phone number to call and was given the contact phone number for the ZZZ Tower. I elected to end the flight immediately and was cleared to return to ZZZ frequency for inbound and landing clearance. Upon landing at ZZZ I was asked by Tower if we were on a training flight (yes) and if there was an instructor on board (yes; myself). I was again told to call Tower immediately upon securing the aircraft. I complied with this instruction and contacted ATC at ZZZ via phone as requested. During that phone conversation with ATC at ZZZ I re-confirmed that it was a training flight; that my student was flying the aircraft; and I was the instructor on board. The ATC controller at ZZZ informed me that there had been an aircraft Approaching final at ZZZ and the proximity of our aircrafts resulted in a 'TCAS' for the arriving Jet. I communicated with the ATC controller at ZZZ that I understood the nature and severity of the situation at hand and continued to express that I was never advised of an inbound aircraft from any controller (either ZZZ before being handed off the ZZZ2; or by ZZZ2 as the aircraft navigated through overlapping airspace inbound to ZZZ). Upon ending the call I spoke with the student pilot soloing in the company aircraft who made his initial inbound call 10 miles from the south and was immediately cleared to land straight in on Runway Y at ZZZ. The student informed me that he remained on the ZZZ Tower frequency for the 10 mile in bound leg at no time was given traffic avoidance for an inbound jet. Per this student's communication he 'was flying straight in for Runway Y (at ZZZ) and all of a sudden was cut off by a Jet' further confirming that 'the Jet just came out of nowhere and ended up right in front of (him) for the runway.' This student reported feeling shaken up and concerned that he also 'never knew the Jet was there until it was right in front of (him).' The student pilot continued to explain and draw out to myself and his primary CFI how he was completely cut off by the incoming jet on a final Approach he was cleared for 8-10 miles out. To further note; the student pilot in company traffic; who remained on the ZZZ frequency communicated to me that the pilot of the inbound Jet told ZZZ ATC that 'it was ZZZ1 Approaches' fault' and communicated that 'they were handed off late.' My student was setting up to join the left downwind for Runway X at ZZZ2 as instructed by their controller. Upon recognizing my student's flight path towards the corner of the left downwind for runway X at ZZZ2 I began verbally instructing a correction; and then taking controls to navigate the plane west bound to 'square' it up when I realized by student was not putting in correct control inputs and rather 'freezing up' on the controls. When given instruction from ATC at ZZZ2 to turn west IMMEDIATELY; I repeated said instruction and executed the command with immediate action. I truly believe the root of problem was the lack of communication between ZZZ1 Approach; ZZZ Tower and with ZZZ2 Tower. I believe the delay in the 'handoff from ZZZ1' as communicated by the Jet Pilots to ZZZ Tower at landing played a significant role and resulted in a lack of; and delay of communication with nearby pilots (myself and student; and a company student pilot on solo). Another significant concern of myself and numerous pilots and instructors based in; and who are familiar with the area is the lack of radar contact at ZZZ2. ZZZ2 is a busy class D airspace which further overlaps shared class D airspace with ZZZ; and falls under the class C airspace of ZZZ1. The lack of radar contact with civilian and military aircraft navigating in and out of ZZZ2 in addition to part 121 traffic; GA traffic; etc.; at ZZZ1 and a significant amount of Part 91 and 135 traffic at ZZZ is; in many opinions; grossly unsafe. I believe if ZZZ2 had radar contact they would have had more awareness of incoming Jet traffic (in communication with ZZZ1 Approach) and could have provided more timely traffic avoidance. In addition another concern; and in my opinion; an ongoing concern for a number of us flying at ZZZ is the delay in which aircraft are 'handed over' by ZZZ1 Approach. On a number of separate occasions while flying in the pattern at ZZZ I have heard the ATC controllers communicate to pattern traffic that 'they have traffic inbound that they aren't talking to yet' because 'ZZZ1 hasn't turned them over' while attempting to re-work inbound traffic for safety and separation. When ZZZ Tower is aware of incoming traffic they will very often hold pilots leaving ZZZ and delay communications with ZZZ2 until the pilot leaving ZZZ has ensured a visual on the incoming traffic. In our case this did not occur as ZZZ did not appear to be in any communication with the incoming Jet. Beyond this; there does not seem to be a true standard practice for safe navigation out of ZZZ and to ZZZ2 for pattern entries. Often as pilots we are told to depart runway Y and execute a 'left downwind departure' which implies that we depart the runway at ZZZ; turn a left crosswind; and left downwind and then continue to fly straight to join a left downwind for Runway X at ZZZ2. Upon contact with ZZZ2 pilots are often told to join the downwind and report midfield. Numerous times pilots may enter at the corner of the downwind for runway X and fly a full downwind without correction from ZZZ2; other times pilots will join at the midfield of runway X without correction or communication for how the execution is expected. *TO note; while departing runway Z at ZZZ2 to enter the downwind for runway XX at ZZZ we are essentially flying towards ZZZ to join a full downwind for Runway XX; essentially making a rectangle. Overall I believe the lack of communication with ZZZ1 Approach; and thus the lack of communication with myself and another student pilot in conjunction with inconsistent expectations and acceptance for pattern entries was the primary cause of the proximity of aircraft that evening (not only with our aircraft and the incoming Jet but also the student pilot in a C-172 and the Jet as he was cut off on a final Approach he was cleared for numerous miles out). In an effort to prevent recurrence it may be helpful to have a standard distance away from ZZZ; ZZZ1; and ZZZ2 in which ZZZ1 Approach should get inbound pilots in touch with the control Tower for which they are landing. In addition it would likely be helpful to have a standard practice for how traffic should enter and exit patterns in and out of ZZZ; ZZZ2; and possibly even ZZZ1 which allows for more consistent traffic avoidance. To note; the possibility of a standardized way for traffic to enter and exit ZZZ/ZZZ2 was suggested by the ATC controller at ZZZ to me during our phone conversation that evening.