A name such as “Robertson” won’t really do because there are scads of Robertsons (Smiths, Joneses, Millers) all over the world. I’m wondering if there are names that really haven’t traveled so much, but are abundant enough in Britain to be recognizable as a common British surname.
Sort of like “Nigel” is for first names–no reason an American or an Australian can’t have it for a first name, but they don’t, generally.
One surname that, while I’m sure could be found in the “colonies” as well, I strongly associate with merry old England, is “Hargreaves”. To a lesser extent, “Higgins”.
First names a bettter clue. A first name is a tipoff as to whether a person is a first- or second generation immigrant.
Spellings like Smythe and Browne remain common in England. A British friend named Parkes told me of of being introduced to an old aristocrat, who exclaimed “With an “E” – Splendid!”
Well if you just want the most common surnames, the Internet says they are:
If you want a surname that strongly ‘suggests’ ‘British’, then you need to decide if you actually mean UK or English / Scottish / Welsh / Northern Ireland.
I reckon that these would ‘suggest’ an origin:
Fotheringay-Smythe
Campbell
Jones
Kelly
There are plenty of Campbells, Jonses, and Kellys all over the world. The OP was asking for names that are mostly restricted to Britain.
That’s difficult because there was so much emigration from Britain in earlier centuries.
Something suggestive - to people in Britain - of a social or regional stereotype. So Bickersdyke or Fosdyke suggests Yorkshire, anything hyphenated (especially if spelt wildly differently from how it’s pronounced -Feathersonehaugh, ffoulkes, Cholmondeley) suggests upper-crust English. There’d be different results for Scotland and Wales (IIRC, in Wales, surnames supplanted patronymics relatively late, which is said to account for the relatively few surnames typically associated with Wales).
Higgins is just as common in Ireland as in England (and other vassal states) so perhaps not so much an English identity as you might think (And I do differentiate strongly between ‘English’ & ‘Irish’).
First Higgins that comes to mind is the current President of the Republic, Micheal D. Higgins, followed closely by the first leader of a fully independent Chilean state, a certain Bernardo O’Higgins …
I was coming in to say this. Thing is, because surnames are passed down and not subject to changing fashions, the most common British names will span the globe.
First names fall in and out of fashion, so Nigel may not travel as well (it’s also well out of fashion in the UK, so you won’t find many toddlers called Nigel).
My surname is firmly English - it has Anglo Saxon origins - but it’s also so uncommon that even British people often think it’s foreign. We’re obviously just not very good at breeding.
Not sure why your english suggestion needs to be so niche-ly upper class, considering the vast majority of English people are anything but - I’ve certainly never met someone with such a fanciful name. Smith, Brown, Johnson or Taylor would be more appropriate, but fail the OPs (in my opinion unachievable) criteria.
I think the OP is asking a lot if we want a surname that is a) common in the UK b) recognizable as a UK surname and c) hasn’t ‘travelled’ abroad…
So Fotheringay-Smythe clearly satisfies b) and c). 
Can’t help this - but how about ‘Featherstonehaugh’, unfortunately not common but uniquely English.
Ask Jeeves.
I tried a google, and it seems the only Fotheringay-Smythes in the world are invented characters - so it has a touch of the Austin Powers about: sounds a bit English, but doesn’t reflect real English people!
Fair enough (I honestly thought it was real 
), so I’ll go for:
Tollemache-Tollemache
which satisfies b) and c).
Huh. Norman interlopers. Coming over here, stealing our lands, beating up innocent Saxons…
Aside from the revelation that Fotheringay-Smythe is a fictional surname (I also had thought I recognized it from a real source…!), the fact is that most or all “common” English surnames are also going to be pretty common in the United States, Australia, etc., as well, due to large scale emigration to these places over the past several hundred years. Which will reflect a lot of Smiths and Coopers, but not so much the Smoot-Hawleys of the world.
(Yes, I know what the name-pair “Smoot-Hawley” really refers to, but come on, it totally sounds British, doesn’t it, at least to an American, in the exact same way as Fotheringay-Smythe, and that’s the point of this thread?)
I’m going to say that a surname of a famous Brit would sound “very British” to Americans, at least, like “Churchill”. I can’t say I’ve ever encountered a person surnamed Churchill in real life, so my only mental image would be of Sir Winston talking about fighting Nazis on the beaches with blood, sweat, and tears.
Good shout - although I guess it sounds typically British only by virtue of a famous Brit. There’s probably other examples we could think of - Shakespeare, anyone? Thackeray? Dickens? Darwin? Constable? Gainsborough?
Thackeray is a pretty good example. It sounds very British to me (not a name I have ever encountered), I can only think of the 19th Century English novelist; and yet it is common enough that there is a pretty good-sized list of “notable” Thackerays in Wikipedia, nearly all British, with two South Africans (plus an Anglicized version of an Indian name propagated by a famous founder):
I guess the difficulty with the OPs question is the ‘common’ bit. Thackeray does sound English, but I don’t think I’ve ever met one.
Regarding Welsh names, not only are English surnames very common there (the most common surname in Wales is Jones; other common names include Williams and Ellis), but so are paired surnames, often a more Welsh-sounding one paired with a more English sounding one. E.g. Ralph Vaughan-Williams (composer), or Sophie Rhys-Jones (wife of Prince Edward, Earl of Wessex).