Well, as this thread seems to have revived itself, I guess I can share my own experiences with the 375 I discussed in my OP in this thread.
Honestly, it was fine. No issues, no delays, no quibbles, I got what I wanted. No more hassle than my application for my 270 or my 44 lever gun. Credit to West Mercia there.
I applied for zeroing and competition use in the UK with a 'good reason' of running boar competitions at the range where I was a member. They were fine with that.
I asked for 100 round ammo allowance so I could reload batches of ammo. They gave me 160.
I specified that I wanted expanding ammo so I could practice with hunting loads ahead of driven boar or trips to Africa. No issues.
2 weeks to turn around the variation, a quick 10 min telephone conversation, job done. Low drama all around.
I didn't particularly want it conditioned for hunting in the UK as I had a perfectly good .270 for that, so I didn't ask for that condition. However, I'm pretty confident that I'd have got it if I wanted it.
Obviously your experience may vary depending on your local police force, but my experience is that getting these rifles, at least with WM, is not nearly as complicated as some people assume it is, or make it out to be.
If you want one, you have good reason and plans in mind to use one, then I say go for it. You'll likely have no issues.
Oh, and shout out to
@bowji john 's safari course mentioned above. If you have an interest, even a passing one, in the whole safari thing, I can strongly recommend his course. It's a lot of fun, and I learnt a lot. Of course, it did lead to me getting my .375, so it wasn't 'cheap' per se, but good value nonetheless!