I can't speak for 637, but 621's history can be traced back to 1943 when it was formed at Weston Super Mud as no. 87 GS and stayed there until they were kicked off by DeSavary in 1993, the longest any School stayed in one place.
Perhaps if you’re talking about a gliding school specifically?
East Midlands UAS was formed at Newton in 1941 (as Nottingham UAS) and remained there until the base closed in 2000, before moving to Cranwell. I can’t think of any other unit that remained in situ for longer.