apostrophe inserted into a three word title

I don't think so Med. But if you want to be technical about its qual as a compound adjective then

We're "Britain's In Bloom" borough

or

We're Britain's-In-Bloom borough

I believe the apostrophe is correct, just because it is the most plausible way to show the posessive. I'm not wedded to the opinion.

They could have written "We are the Britain in Bloom borough.", but that just evades your question.

I wish the OP had mentioned which borough it was - there are a number of categories of awards in the Royal Horticultural Society's "Britain In Bloom" competition, ranging from small villages to large cities. The 2013 final results will be awarded in October.

The competition title seems usually to be treated as a fixed phrase, as the OP recognises. I see that Newcastle-Under-Lyme in Staffordshire held a "Britain In Bloom borough painting competition". I can imagine that the winner of the "borough" category might brag that they were (e.g.) "Britain In Bloom borough 2012", except that there isn't a "borough" category.

I can imagine a community saying e.g. that they were "Britain's In Bloom village" if they were the current year's winner of that category.

More info please!

Perhaps the indefinite article 'a' is left understood. Am I correct in assuming that they are not the only one?

'We're a Britain's In Bloom borough'

Well, the apostrophe would be an error, given that the competition is "Britain In Bloom". Each year there are plenty of finalists. In each of these categories there are Gold, Silver Gilt, Silver and Bronze winners, plus a Champion Of Champions: Small Village, Village, Large Village, Small Town, Town, Large Town, Small City, City, Large City, Urban Community, Small Coastal (electoral roll up to 12K), Large Coastal (electoral roll over 12K)

It isn't grammatically incorrect if they mean to say that they're Britain's "In Bloom" borough. It looks like they're making a play on the title "Britain In Bloom." It would be more percise, but also more clunky, had they said "Britain's Britain In Bloom borough."

I must say that British municipal employees are not necessarily recruited for their English language skills. I once noticed a new, very large white enamelled steel notice, with raised black lettering, in a precinct near the bus station in Bristol, announcing, among other things, that the public toilets were "closed on Sunday's". I wrote a jokey letter to the council saying that newly arrived travellers from other towns would think that Bristolians were all thickos and suggesting that an employee should go there with some white paint and a brush and white out the offending apostrophe. After a couple of weeks I got a letter back thanking me for pointing it out, and not long after I saw that my idea had been adopted.

Oops Med, again didn't read carefully, just now realize you were asking about the possessive. Yes, no, like the other guys said, it doesn't look right

However I might have written,

We're Britain In Bloom's borough

or even

We're Britain In Bloom's borough

Though yes, no, probably nobody else would do it