Back in the 60's, most UK recording acts were looking for potential hit material to help launch / sustain their careers.
UK record company A&R guys would get hold of copies of all new chart entry US 45's to 'find' potential songs for their labels acts to cut.
When one 'likely tune' was come across, the task was to get one of their acts into the studio instantly, cut the song in a day (many times just copying the US original as closely as poss) and get it mastered, pressed & into the shops asap.
That way they could 'beat' the original recording onto UK radio & hopefully get the British hit version.
Don't forget that pre-pirate radio, most UK record spinning radio shows were told to play a very high proportion of Brit recordings than US.
So they could play the UK cover but not (usually) the US original version.
Pirate radio changed all that.
Acts such as Georgie Fame, Dusty, etc loved US soul tracks and would cut their versions to pay homage to the original version.
Many times, their UK label A&R guys had diffferent ideas and would select a particular track as a UK 45 A side when the artists just wanted them as LP tracks.
Georgie Fame had a major bust-up with his UK label coz they kept putting out his' tribute covers' as 45 tracks when he hadn't wanted this to happen.
Some times a UK label would even cut a Brit act on a song when they held the rights to release the US original anyway. It wasn't unknown for them to put both versions out here.
So, I'd say it was more down to the label's A&R men than to a newly signed act what 'cover tracks' they cut.