The only way one could argue UK does not have nationalism is to adhere to a where narrow definition of nationalism that not really mirrors the way it is used in daily tongue. If one defines "nationalism" as in "wanting to unite/combine several more or less similar cultures into one for the 'greaterment of all [people of that culture]", whether forcefully or not, then yeah, it could be said that Britain doesn't have nationalism, given that it's more of "unionism" in that way - keeping several more or less different cultures tied together in one state.
The thing is though Scriver a number of parties want to do exactly that. Basically every mainstream Westminster political party, in fact. Granted they're happy to string people together as part of a multicultural, multinational union but that doesn't really explain Labour's new slogan "One Nation". The Westminster parties in relation to the independence debate are all about uniting our cultures into one, though allowing us to remain autonomous ("a strong Scotland in a United Kingdom" etc).
That said, in common British parlance "British Nationalism" tends to be along ethnic lines and it tends to be English in nature; these groups just lump all "British" people, whether they're Scots or Welsh or English or Northern Irish, into one big group of White People standing in opposition to the hordes of unsightly brown people ruining England's green and pleasant land with their Islamic barbarity. See
the British National Party,
UKIP,
the National Front,
English Defence League,
British Union of Fascists and so on. These parties are more about "shipping immigrants back" than anything else and keeping Britain white and ethnically "British" (even though there is no such thing).
These groups don't tend to do so well outside of parts of ethnically divided England, though in Scotland and Northern Ireland the small gains they have made have been for different reasons. This is where the other manifestation of British nationalism comes into play. In Northern Ireland and Scotland, British nationalism is what crops up as a way of expressing hatred of Irish Catholics, or in a more general sense opposition to the Republic of Ireland's claim in Northern Ireland. The case for Northern Ireland being a British province is generally founded in a sense of "British solidarity" among certain Protestant Northern Irishmen who claim Lowland Scottish ancestry, but their Britishness only goes so far I feel. If you forgive my crude generalisation, the Northern Irish hate the English as much as Scots are alleged to. It's more about opposing Catholicism and the RoI than anything else.
On the few occasions that the two strands of British nationalism interact with one another they tend to find that for some reason anti-Irishness, Sectarianism, Conservatism/Right-Wing ideology, British Unionism (which is usually a very nakedly nationalist ideology) and White Nationalism go hand in hand, leading to the creation of groups like the
Scottish Defence League or the National Front in Scotland. Rabid Rangers Fans, in other words. This is why bigotry plagues British Unionist groups in Scotland far more than it does the pro-independence groups. You might hear the odd independence supporter with more keys on his keyboard than brain cells telling David Cameron to "fuck off back to England" but that's nothing compared to what you'll hear from the far-right groups campaigning for a No vote. Fringe they may be, yes, but as far as fringes go they're pretty awful compared to our fringe groups. The worst we've got are Gaelic-speaking Communists.