A Tower Controller reported they turned an aircraft off of the final approach course due to being overtaken by traffic on TRACON frequency. TRACON turned their faster aircraft too resulting in an NMAC.

Narrative

Aircraft X contacted the Tower about seven miles northwest of the field at 2200 ft. and was tagged by the Approach Control for the right downwind entry to Runway XX; which was the runway in use. The entry to the downwind was inefficient for the pilot since they were coming from the north and added workload to the Tower that seemed unnecessary. I instructed the aircraft to make a straight in entry to Runway XX. I scanned the final to make sure it would not impact the approach. The only other aircraft was Aircraft Y estimated about 13 to 14 miles north for an instrument VFR approach. As Aircraft X continued inbound; they descended to 1700 ft. at about 80 to 90 kts. Aircraft Y was observed at 180 kts. and no slowing down. The separation on final quickly reduced to 3 miles and then even less. When I realized that Aircraft Y was not slowing down; I was not in contact with the pilot yet. I instructed Aircraft X to make a turn southwest and plan for a right downwind entry because of the faster traffic not slowing down behind him. Aircraft Y still maintained 180 kts. at around 5 to 6 mile final; so I instructed Aircraft X to turn westbound to ensure they were clear of final for the fast moving aircraft. Aircraft Y was observed breaking off of final at 2000 ft.; westbound in the same direction as Aircraft X. Aircraft Y passed just above and behind of Aircraft X at a fast rate of speed. I did not have a chance to issue the traffic to Aircraft X since the move of Aircraft Y was sudden and unexpected; also the Aircraft Y was behind Aircraft X. The pilot of Aircraft X spotted Aircraft Y after the aircraft was safely ahead and to the right.A recent change has been made to allow Approach to put single engine VFR aircraft into the midfield downwinds without coordination to allow more efficient sequencing of aircraft and alleviate some workload for Approach. The fact is that Approach allows aircraft into runway departure corridors and ships aircraft that should be sequenced straight in to the runway and do not instruct aircraft to enter the midfield downwind; nor do they vector them to a point to allow for that to happen.Since the change numerous aircraft have been observed entering the departure corridors at 2200 ft. on heading usable by the Tower to put Jets on; which climb to 3000 ft. Also aircraft; like on the Runway XX operation are switched to Tower north of the field tagged as left and right downwinds because it easier to get rid of them; but they should be sequenced with their traffic. The new downwind procedure should not happen until there is a clear understanding.