Yes, there is. I simply can't see how a group of people living together does not eventually adopt their own traditions and mannerisms.
Britain does have a culture, it's just harder to define because its four nations in one. I mean, how would use describe the culture of a hyperthetical Federal Europe? There would be some kind of similarities between the different european nations-or states, but largely the cultral differences would be within the ex-European states. That means whilst a Federal Europe won't have a culture per se, because they aern't one people, it would still havea national culture, just its culture would be made up of many different kinds of cultures. Such a system is common in African countries. Where many tribes and ethnic groups have their own strong culture and as such "Nigerian culture" may apply to which ever localised culture the particular individual invoking Nigerian culture may belong to.
The trouble with British culture is, in the past 30 years or so we have downplayed the importance of English culture. Why-I don't know, but perhaps it was an attempt to psychologically incorporate the rest of Britain into national thinking. For example, abroad, people think "British accents" are those of Stephen Fry or Jeremy Irons, when these are simply English accents. "British food" is often mistaken for particuarly English dishes. Yet it has become unsavoury in recent years to think proudly as oneself as English as a Soct would consider themself a Scot. I believe this has fuelled the rise of Scottish nationalism in recent years. If English culture is now transplanted onto British culture, then by asking the Scots to be British, we are asking them to become English. What it means to be British, in my view, is the belief as an Englishman, a Scotsman, Welshman or Northern Irishman, these four different peoples are the same as your own, despite being cultrally, and in some cases linguistically different. By constructing a Britishness, mainly from the government which represents England, we risk pushing the other three nations of the UK away because that's not what true Britishness is per se.
So to conclude, yes Britain has a national culture. It's just complicated and diverse. More importantly, England has a rich culture, but it is the fusion of the four nations that make an English culture. I am sure there are personality similarities between the four UK nations that would differ say, to continental nations, and of course there is the overarching similarities such as language, faith, views etc. But generally speaking, I think the United Kingdom has a great culture and we should be happy of it.