The better question to ask is: Does Britain have identity as a nation?
Various surveys indicates that a lot of people in the UK - across all the constituents parts - see themselves as more English/Welsh/Scottish/Irish than British. Many don't see themselves as British at all (ironically immigrants who have become citizens are far more likely to see themselves as British first than people born here).
The British identity as a single nation is weak and fractured.
Keep in mind that the modern nation state and national identities are barely two centuries old and was largely an artificial construct to begin with, created as part of the romantic nationalism movement, and it has had various degrees of success in different places. In the UK a lot of that identity got built around and tied up in the Empire, and has fractured and gone into decline with the fall of the British Empire.
On top of that, I'd argue that London in many ways does have a distinct culture from the surrounding area. London - especially the centre - is a Labour stronghold in a sea of conservative areas. It voted far more for Remain than the surrounding area, despite a large proportion of the residents of London being unable to vote (seeing as London has a high proportion of non-citizen residents). It's far more immigration friendly, far more multicultural than the surrounding areas.
After the Brexit there has been proposals - some serious, and some not so much - for an independence movement for London. I know some concrete planning has been done. There was real and seething anger in London over the vote - in many areas it was seen as outright betrayal, and evidence that London does not really fit in with most of England. People joked about building a wall around the M25 (freeway circling London).
Now, there's no way there's any sizeable real support for an independent London, but the Brexit vote did plant a seed. We'll see if it grows - if the terms we leave the EU on are bad, and especially if they hit London hard, it very well might start growing. I genuinely think that while it won't get much support, ther likely will be a registered political party arguing for an independent London within the next decade.
Britain is not a nation, it is an island, so the question doesn't really make sense to ask. It's impossible to be English, Welsh or Scottish and not be British; you can't choose geography.
You're being a pedant and intentionally obtuse, so let me be one too: While we often refer to the island as Britain, it's name has been Great Britain for many centuries.
The use of variations of Great Britain for the island dates at least as far back as Ptolemy - the use of variations of Britain alone (without some variation of "Great") quickly fell out of use around that time.
And as I'm sure you're perfectly aware, Britain is equally used to refer to the UK in common parlance. The history of using precursors of Britain (without "Great") to refer to a political entity dates back to the Romans, who started using Britannia to refer to the Roman province.
It was exceedingly clear from context that it was used to refer to the UK rather than the island. When talking about national identity, one also specifically talk about "British", which makes it more natural to use Britain rather than the UK.
