I dislocated my middle finger a few years ago. Like you I snapped it back in straight away and had it x-rayed at a walk in centre, with no obvious break found.
A month later I was still in a lot of pain and couldn’t bend it, so decided to get a second opinion. Got a referral from my GP to see an orthopaedic specialist, had another X-ray to find when the dislocation occurred a sliver of bone had been pulled away with the ligament.
After physio exercises and months in a splint, pain went and got somewhere close to full movement back. Still can’t quite extend it fully, but functions okay...not sure whether the delay in treatment impacted on the recovery.