Sometimes abbreviated to best of British. I had wondered whether this (especially the shortened version) would be understood in the US. I see from the OED that the earliest cite is 1960 from an American newspaper but I’m loth to draw any conclusion about usage in the US from that as that whole sentence seems rather out of place for an American; I’m guessing they’re quoting a British speaker. There is the David Foster Wallace cite but as it’s from a book of short fictional pieces I’m not sure of the context or character speaking.

Here’s OED:

(the) best of British luck: an expression of encouragement, often with the ironical implication that good luck will not be forthcoming. Also best of British.

1960 Tucson (Arizona) Daily Citizen 12 Nov. 44/3 The very best of British luck to the great little destroyer and all the gallant kindly company of men who are taking the White Ensign to the other side of the world. 1963 L. MEYNELL Virgin Luck ii. 39 As soon as they [sc. kittens and young birds] can fend for themselves they get shoved out into the world and the best of British luck to them. 1967 W. KEENAN Lonely Beat vii. 74 ‘Best of British.’ With that the reporter vanished. 1999 D. F. WALLACE Octet in Brief Interviews with Hideous Men 146 Right from the start you’d imagined the series as an octet or octocyle, though best of British luck explaining to anyone why.

Two questions then. Would best of British be generally understood in the States as wishing you luck, and, if so, would it have that same ironical undertone?

This American has never heard it before.  Without plenty of context, I would not recognize the truncated form as referring to luck; in the full form, I think I would have assumed a certain level of irony (nobody talks about “the luck of the British") but would probably have inferred it was actually wishing ill luck.

Not heard in my corner of Leftpondia either.

The quote from the Tuscon Daily Citizen seems to be referencing Briish ships (White Ensign), so the full phrase would likely be put in context by the rest of the article and be understood by readers to mean simply “good luck to the Brits”.

Never heard of it, wouldn’t have understood it.

Another nevah-hoid-of-it here.

I’ve definitely heard it but can’t remember where, or which class used it. There was a big “I’m Backing Britain” campaign in the ‘60s and we schoolkids were given small Union Jack stickers we plastered everywhere probably when the government began to realise Japanese motorbikes were vastly superior to BSAs, Triumphs, Nortons, etc.  not to mention tape decks and cameras and film. It could have revived as an expression as a response to this but, either way, it now smacks to me of the “you’re a white man” kind of blinkered ‘compliment’ said when someone does you a kind turn or a favour.
Was “you’re a white man” ever used in the States?

No, but “that’s mighty white of you” was (is?) unfortunately common.

it now smacks to me of the “you’re a white man” kind of blinkered ‘compliment’ said when someone does you a kind turn or a favour.

It doesn’t convey that to me. My family have always used it, meaning “fine, go ahead, but I don’t remotely expect you/them/it will succeed”. What it smacks of to me is the 1950s black-and-white British war films in which a few of our brave lads in khaki prepare to dig in and hold the line against overwhelming numbers of much-better-tailored Germans.

Once again a quick Google Books search finds a number of theories, notably Eric Partridge (yes, yes, I know), whose 1986 edition of A Dictionary of Catch Phrases: British and American, with Paul Beale, dates it indeed to the Second World War, and quotes the comedian Frankie Howerd, who claims to have popularised the expression on the 1950s radio programme Variety Bandbox.

Google Books claims to find the phrase in the pages of Hansard (the official record of debates in the House of Commons) for 1937, but it’s “no preview available “ (why, for crissakes?) and I don’t think the “official” version of Hansard has been digitised that far back yet ...

Aldi, I’ll help you out by saying that I have heard it before. I’m in Oz, so that might make a difference. My mum still says it (78 year old).
I would doubt it has much modern currency. “Best of British luck” would be understood as wishing you the best, even if the British reference was not understood. “Best of British” will simply flummox most Aussies now.

I’ve heard it on TV, never encountered it in the wild.

Not familiar to me.

But G. Books search turns up many examples of “British luck”, sometimes in the extended form “British luck and pluck”, often referring to the good fortune of the Empire, since the late 19th century: e.g.:

1894: // Of course you will trust in British luck and pluck; ....//

1898: //the advocates of British luck declare her navy invincible ....//

1899: //nothing but a double dose of British luck and pluck saves us from disaster ....//

1901: //With true British luck, Port Arthur may silt up, or the Russian Empire may break down.//

1902: //the government has a sublime confidence that British luck and pluck will save the country ....//

1906: //The proverbial “British luck,” however, has established the empire ....//

1908: //The world-famous British luck is another name for British illogic.//

1918: //But British luck or pluck might pull us through.//

I suppose it’s natural that we Brits considered ourselves lucky in the days of imperial supremacy. I wonder if the ironic use of the phrase came with our fall from dominance, although I’m probably over-analysing. Language doesn’t usually work like that.

I suspect that it just became such a cliche that eventually people no longer took it seriously enough to use it other than in an ironic sense.

As the Venomous one said, could well be a 60’s public info campaign of some sort. The alliteration involved might support that theory…

...but it sounds more like something from the Royal Air Force types during the Battle of Britain in WW2. One of those stiff upper lip things mentioned to cannon-fodder bomber pilots heading off to the Scandinavian fjords or Ruhrgebied industrial targets.

But I CBA checking it out..., sorry, other thread there… off to my kip now, smoke me a kipper for breakfast!