"So what does the UK gain by dangerously escalating the war with Russia? It seems almost desperate. Why else would you push for such risks? What do they stand to win or lose?"

What they often get are the crumbs from the masters table...though usually they try for a bone.
It's one of those things hard to work out perhaps because it reflects multi-dimentional UK stupidities conceits and delusions.

Lots of the elite think that they are "the same" as the Americans ( "they" speak "our" English after all) and vice versa, so they don't see the reason for a differing UK policy. The US meanwhile view the UK more as an expendable appendage, but there's a masochistic subservient UK element in the "special relationship" : a nation of cap doffers who "know their place" are willing to play denial on that one. They also think of the US as carrying on the great UK empire "tradition": a kind of "Our Greece to their Rome"...and are thus keen on the US continuing with their own inherited Russiaphobic ideals and policies.

Of course a little country well past its Empire " sell-by" date should be focussing on its own future security stability & wealth but alas, delusions of grandeur are all too common in the UK elite and it's military establishment.

They will always win out over mundane realities.