Was it better or worse than Julia Louis Dreyfus’?
Neither of those gentlemen ring any bells. What might I have seen them in?
Magnum PI and The Nanny, respectively. (Davis was also Professor Moriarty, on TNG.)
I win the thread!
Ah, it’s been many, many years since I saw Magnum PI. I don’t think I knew the guy’s name then, either.
Never heard of The Nanny, I’m afraid.
But even if I had heard of 'em, I’d still stand by my “can’t think of any offhand” - the OP was asking about “Brittish or Austrailian TV shows” (whatever a “Brittish” or an “Austrailian” is; probably something like an Ammerician).
The Iron Lady - British
Shakespeare in Love - British/American production: British/US writers, British Director, US Producer
Bridget Jones - British
For what it’s worth, Davis was also in The Prestige – which arguably fits the OP’s “films” criteria – as the proper British judge dealing with Michael Caine at the outset.
I can recall some Brits refusing to believe that that was not their native accent.
You only have “person” so far! You said “people”! Premature declaration of thread winning!
And, upon checking IMDB, I see that Hillerman played John Watson – to Edward Woodward as Sherlock Holmes – in a British television movie, “Hands of a Murderer.”
What about the “great” Jonathan Harris (Dr. Smith from Lost in Space)? Maybe he falls into the Hillerman category?
How about Vincent Price?
Vincent Price used a British accent?
I think Theatre of Blood counts.
I guess Irish accents are technically non-UK, but Brad Pitt in Snatch?
The OP asks for suggestions from British TV programmes and gets almost entirely suggestions from British movies and American TV shows. 
Perhaps that’s just an indication of how few Americans there are being British on British TV shows. I guess there’s enough TV being made in the US that there’s no need for them to move.
I can’t actually think of any except the aforementioned Gillian Anderson (Great Expectations and a couple of other British mini-series that I haven’t seen), if she even counts.
We had a thread a while before about that bloke from Magnum PI; I was one of many who hadn’t even realised he was supposed to be English. He had that peculiar faux-English accent that a lot of American actors used to have (British actors have a similar faux-American one, particularly in radio plays), which doesn’t actually sound like an English accent at all but does enunciate the Ts and drop the Rs.
Why? Northern Ireland was part of the UK last I checked.
Dick Van Dyke did excellent accent work in Mary Poppins.
The senior Mr Dawes.
That wasn’t a Northern Irish accent.
Continuing the hijack, having first encountered Adrian Lester in Primary Colors I was surprised to discover that he was British. Emma Thompson’s accent was, alas, not as good.
I will also mention that UK television shows do feature a lot of British actors doing appalling American accents (Spooks, I’m looking at you here). If there are any American actors doing good British accents on UK television they’re doing them so well I haven’t noticed.