Business jet Captain reported an NMAC during descent into TEB. Crew followed guidance and continued uneventfully.

Narrative

During the descent from 6;000' to 5;000' whilst on track approximately due east to the STRAD intersection in preparation for the RNAV (GPS) Y Rwy 19 approach at TEB; and working New York Approach on VHF 127.6; we received at TCAS [descend] RA. The Autopilot was engaged in modes NAV [FMS] and VS. The Autothrottle was also engaged with a command speed of 250Kt. At no time were we offered any traffic information from ATC. The RA was immediately followed by autopilot disconnection and moderately aggressive nose-down deviation. The TCAS 'intruder' was visually acquired quite quickly; but it was initially difficult to assess weather the aircraft was headed toward or away from us. The FO provided verbal interpretation from the instruments whilst I concentrated on flying the aircraft and looking out at how the situation was unfolding.I decided that a right turn in addition to the descent was prudent and executed. Despite a relatively quiet environment; it took several very clear assertions to ATC about the situation we were in to get an acknowledgement; which was nonchalant at best; and simply concerned what to do when we were clear of conflict.We passed below and to the right of Aircraft Y. We were overtaking the Aircraft Y very quickly; and given the lack of extra visual references (I was looking directly at it against a blue sky and nothing else); it was so close that I feel I confident could've have read the N-number had I not been rather preoccupied with other endeavors. We reached approximately 4600' and less than 1/4 scale right of course in the deviation.Following the TCAS RA we inquired weather Aircraft Y had been visible to ATC; who informed us that the traffic had 'Just popped up' and was 'at 5;600 and not talking to anyone' we found this unfeasible given that; to interact with the equipment in Aircraft X; Aircraft Y must have had an operating and altitude reporting transponder.After the 'Clear of Conflict' call; the controller began to give us an approach clearance to which we responded 'Aircraft X Standby.' This was because we had further traffic converging from the right; roughly perpendicular to our track; just 100' below us. The controller responded by immediately reiterating the clearance in a rather elevated tone; leading us to explain the situation. We never did visually acquire the second aircraft and although it had us concerned no alert was triggered.Whilst it took several attempts to have the clearance re-read to us after the second traffic had passed behind us; the rest of the flight was uneventful.Cause: 1. The issuance to us of a descent clearance; through the level a much slower VFR aircraft on an identical track.2. The lack of any; let alone timely passage of traffic information by ATC. This was compounded by the lack of angular velocity between the aircraft; and that the aircraft were nose to tail making visual acquisition that much more problematic.3. The decision of the other crew to operate VFR in the vicinity of busy airports and their associated procedures compounded by the decision to do so without talking to a relevant ATC facility.Suggestions: 1. Timely passage of traffic information by ATC; and the avoidance of clearances likely to cause a conflict: If; instead of to simply descend 5000; the clearance had been 'Aircraft X; turn right 10 degrees; then descend 5000; When level direct to STRAD; be advised there is traffic not under my control in your 12 o'clock at 5600 not under my control' this would NEVER have become a conversation. The airspace was not busy; I see no reason this shouldn't have been the case.2. Address ATC potential Fatigue/Staffing and Training issues: We know that part of the reason for delays to our schedule later in the day was ATC staffing since this was stated by one of the clearance controllers at TEB.3. Possible revision of current; or provision of additional VFR charting clearly showing which ATC facility best to contact in the vicinity of busy airspace where such contact is not mandatory; and depicting busy IFR routes in that airspace.