Not sure what "control" means.
First, let me say I'm very displeased by this deal. Nobody is watching LIV,, the defections are over, and they weren't a threat to the PGA. The suspensions haven't been lifted. It was a need money grab an wasn't necessary but I don't see how they control the PGA.
As to British golf. This merger involves the DP World Tour. That tour covers European & UK events. So, British golf is already part of this.
If the control actually exists, then yes, British golf is controlled too.
,

You can no longer call it a "gentleman's game" when you hand it over to people who like to dismember people that don't agree with them. Period.

...I'm very displeased over this arrangement.
The question in the OP was about British golf being above or immune to it.
They are already part of it. They can't be immune.

No offense offered nor intended
But you sent me into confusion land

The defections are over and they
Weren't ever a threat to the PGA

The suspensions haven't been lifted
And in the end it was all a money grab

It wasn't really necessary and now
The Saudis may rightly control the PGA

This merger involves the DP World Tour
And that tour covers European and UK events

So British golf is already part of this
And just maybe they are also controlled by LIV

...before I jump to that conclusion.
I think it more likely that Saudis are using golf to sanitize their image (as if) and perhaps clean dirty money.
So far, nothing suggests they're running anything. And, the deal has been struck, so if they take their ball & go home, they'll get sued.
Fact is, LIV was hemorrhaging money. Ratings for 3 rounds here on Chicago TV were 30% of Thursdays on Golf Channel for the Tour event. They're not selling out at the gate either, despite the lower ticket price.
The Saudis admitted there was a limit to the cash flow, and this merger gave them more for their money.
Their attempt to clean their image is polishing a turd.

Of course, it's hard to find anyone that's truly patriotic in this country instead of just flag waving and cheering.

Especially given that the players loyal to the PGA Tour now have victim status, as they weren't even asked what they thought about this.
Golf fans are not going to blame the players, and that's all we care about watching. Who the commissioner is doesn't matter at all, and nobody cares about business matters of the tour.
It was an unnecessary move & I can only believe that, either; Monahan is being greatly compensated, or he's an idiot who made a merger the tour didn't need.
I see no third option.

The PGA players could start another Tour but that would cut into their revenue.

TV contracts, sponsorships, venues are already contracted until 2027.
They could harm the Tour, but there'd be no tour for them.
Honestly, the only way it would work is if Tiger was healthy & he, Roy, & Ricky set it up and all promised to play every week.
The networks without golf contracts would stand in line to air it & every golf equipment company, bank & financial group would find the money to buy ad time.
But, Tiger can't play once every 2 months, and even Rory is no Tiger.
The market just isn't there.

He "capitulated" to an opponent they were already crushing.
Naked money grab.

You have the tail wagging the dog.