Pilot reported an NMAC while entering the traffic pattern at an airport with an operating control tower. Tower controller vectored the reporter's aircraft away from traffic.

Narrative

I am based out of ZZZ airport; which is in close proximity (6 nm) to ZZZ1. Both airports are Class D's under the ZZZ2 Class C airspace; which is 7 nm to the NE of ZZZ airport. Needless to say; it is a busy airspace.On Day 0; the wind was out of the NE at about 10 kts; with ceilings 2300 broken when I copied the ATIS information at the start of my flight.I had a simple mission -- go to ZZZ1 airport for 3 consecutive Power Off 180 landings and then return to ZZZ airport. I warmed up with a short field landing; and upon doing 3 successful accuracy landings I called ZZZ1 Tower (on the downwind after a touch and go) to ask for one more touch and go before returning to ZZZ airport.I was cleared for the touch and go and after to proceed direct ZZZ airport. ATC then got fairly busy coordinating a VFR approach > 1 lap in the pattern > VFR departure to IFR for another aircraft.I did my touch and go; took off; and climbed to 1850 which is the TPA (Traffic Pattern Altitude) for ZZZ airport. I told ZZZ1 tower that I had ZZZ airport in sight and they told me fly direct to the numbers at ZZZ and contact tower. I switched freq's; called ZZZ Tower and told them I was off ZZZ1 airport; inbound landing full stop.Tower immediately vectored me to the northeast for traffic. I looked down at my EFB traffic map on my yoke-mount (using onboard ADS-B-In) and realized that I was basically at the same altitude and location as another aircraft. I made the left turn NE. There was an aircraft on the downwind at ZZZ airport and another aircraft; Aircraft Y; on final. Aircraft Y is the one that I almost collided with. They were on the ILS for Runway XX. I landed with no issue.There is a procedure for this very situation; but ZZZ1 tower does not follow it and does not have radar. I have gathered that communication between ZZZ2 Approach; ZZZ Tower; and ZZZ1 Tower is not always perfect. This highlights one of those instances.A recommendation states an immediate east or west departure when departing ZZZ1 Rwy XX due to ZZZ airport ILS runway XX traffic on final.On a flight on Day 2; we did our landings at ZZZ1 airport and requested a left downwind departure heading 270 intentions return ZZZ airport. It took an extra two minutes and was safer and far less rushed. I believe the school may implement this procedure as a school policy.Perhaps this could happen officially. ZZZ1 tower also does not have radar; so that could be another solution.