Correcting the speed of American programmes

It looks like film, the end credits mention "Panavision cameras and lenses"- Panavision were entirely film and didn't do video cameras in the 90s, I don't think (a bit of research suggests their first non-film camera wasn't until the mid-00s)? And Wikipedia can be edited by anyone, so can't be relied on as the most accurate source.

I know many of the US sitcoms of the late 90s and early 00s which were shot on video and filmised often ended up going out over here without the filmising (and there was one case, Taina, which Nickelodeon showed the filmised version but Channel 4 the non-filmised one), possibly because the standards conversions would make it look a mess? For some reason video-look sitcoms seemed to largely disappear in the US after the mid-90s, whereas we still make some sitcoms that way.

Actually, a thought, with the OP saying they thought the Fresh Prince on Sky Comedy sounded sped up... they aren't using these versions as well, are they? That would explain it.

Could be that the distributor decided that the best "universal" format was 24fps and that's what Sky and the BBC are using, being the highest quality source. I'm not sure what streaming-service support is like for 50 or 60fps. I remember series 5 of Doctor Who looking like 24fps when I played it from Netflix via my STB (which was fixed to 50fps).

From my recollection 'projection speed' has changed plenty over the years to match what was shot. In the UK 16mm was changed from 18fps to 24fps to gain a smoother look (even 9fps was used much earlier).
Then along came TV and to eliminate flicker 25fps was selected as it matched the electronic cycles which make up our 240volts.
The same was applied to the US but as their voltage is either 110 or 120v, 30fps was used.
That was all based on film of course then along came video...
Then computers...
Modern phones and computer editing software is often set to 30fps because of Hollywood/US influence which is fine on your screen, but as soon as it's transmitted on a UK TV programme - unless it's been adjusted - will flicker like hell.
That will also explain why older feature films shot the US might have a noticeable pitch change on UK TVs. Standards conversion is better now but earlier NTSC to PAL were pretty ropey with weird flesh tones and stuttering camera moves.
The same also now applies to stuff shot on 4K images and edited into UK HD and SD broadcasts - it can vary in output depending on how it's used.

Only iPlayer has 50p support I think, and even then only for HD or upscaled HD - the standard SD uploads are 25p only. Fresh Prince has always existed on really grotty standards conversions in PAL territories - and none of those artefacts can be seen on the current iPlayer version. Its been uploaded as 720p50, but is actually 25p with repeated frames - not a 59.94>50 conversion. So a sped up 24fps version would appear to have happened.