It's true, mostly. You really need a certain level of lifestyle to go shooting deer in UK. Scotland, of course, IS part of the island of great Britain, and has far more deer-hunting- stalking it's called here - than the rest of UK combined. Remember that it is also by far the least-populated part of the UK - by a factor of around at least ten to one.
Let's say that the US shooter is after a decent-sized whitetail.
In the USA, most everybody has legal access to a suitable rifle of one kind of another to go shoot a deer. Even a hand-gun, TBH, if they REALLY felt their wood- craft skills were up to it.
And let's assume for the moment that we are talking about BP muzzleloading-type shooting.
They walk into a store in any of the states where a BP rifle is not a even classed as a firearm, buy it along with the necessary supplies, draw a tag for the ML deer season, and go shoot it. Paperwork? A state tag, and maybe a hunting license, just maybe.
How would they do it it UK?
Simple.
They wouldn't.
A number of us here who live in UK have explained, quite often, it seems, that in UK you can't go shooting live game bigger than a rabbit with a black-powder shotgun, but no black-powder m/l rifle is deemed suitable to take anything bigger. And since you can only have a rifled firearm for 'good reason' and deer-stalking is not an acceptable 'good reason' because of the perceived lack of oomph inherent in ANY muzzleloading rifle, you are, not to put to fine a point on it, stuck.
As for unmentionable rifles being used, sure. Youtube is loaded with articles on deer-shooting/stalking here in UK.