Director Richard Lester claimed that he shot at 25 frames per second, rather than standard 24, so as to match the scan rate of the PAL (European standard) TV monitors that are seen clearly in the studio scenes. If so, playing it back at 24 fps would slow down everything by 4 per cent.
How can that be true, if film is traditionally 24fps worldwide, and that theaters around the world always use the 24fps prints? Also, Ron Furmanek, who was making the restored soundtrack in 1994, claims that the film uses the recordings at the right speed, and that all the mixes for records were sped up to sound more lively (Most, if not all Beatles songs, were sped-up from their original speed for release on records).
I'm not following the first sentence of your second paragraph. The fact that film is traditionally shot at 24 fps, and that theaters all use 24 fps is exactly why Lester's story could be true. It's why the movie was projected at 24 fps, rather than 25.
Movie cameras, at least some of them, could be set up to shoot at 25 fps. My understanding is that this was fairly common in Britain when shooting film to be shown on TV, since European TV was at 50 fps interlaced. Since Lester had worked in TV, the cameras he had access to may well have been set up to shoot at 25 fps. It's possible he did it by accident, and invented the sync-with-video-monitors story as an excuse.
As for the musical recordings, we just don't know. The movie could have started with the sped-up recordings, and slowed them back down to the original tempo (which would be a bit of a coincidence). It's also possible that whoever dubbed the music onto the film did it with the film at 24 fps. Just because it was shot at 25 fps (if it was) doesn't mean everything was done at 25 fps.
In the US, the video and film rates are sufficiently different that you couldn't mix them without the result being really obvious. If you were to shoot something at 30 fps, then show it at 24, it would look and sound obviously "off."
Actually, the original poster is confusing Lester describing how they shot the television sequences and the telecine process.
Lester recorded the sequences where a television monitor was in the screen at a frame rate of 25fps because Europe uses 50hz for there monitors thus the frame rate for a film camera needs to be a ratio of 50 (or 25fps) to avoid the pull-down/flicker in television. Then they correct the sound during the final editing.
Any time a television monitor is in film the cinematographer must adjust the frame rate slightly to compensate for the screen refresh rate (either 60 US or 50 for Europe).