I think most players in game (but also in real life i guess) take a second European nationality to count as an EU citizen and thus (in many leagues, like spain and italy) not as a foreign player.
Some leagues have restrictions on the amount of foreign players in a match day squad or even in a squad at all (spain for example). So by taking the European nationality, "Non EU"-players remove that restricition from themselves and are therefore a lot more interesting for clubs in countries that apply these restricitions.
This is the reason many South Americans trace their family lines for an European ancestor and why South Americans in Spain always take up a Spanish second nationality when they have the chance (unless they already have a different European second nationality; Italian is a pretty common one).
EU players who apply for British citizinship (at least in game) have another reason: they do this to be eligeble to play for England. This seems a little buggy in game if you ask me; I have seen (Dutch) players apply for a British passport and stating that they wanted to play for England, while they were already capped for the u21 side and therefore declared for another nation already.
Another bug connected to this is that Dutch players, who are declared for Holland but have a British passport and 'wanted to play for England' when they applied, are still scouted by the England manager 'for the next England squad' every once and a while. Even a player I have capped for Holland myself (I am their manager in game) is still considered for England on a regular basis....
I noticed that Peter Lovenkrands is eligible for a British nationality on days however next to it it says he hardly speaks any English. I assume this means he can't get one. Well from interviews that I have heard He seems to know English really well. In fact he even speaks it in a sort of weegie accent after his time at rangers.
If you listen to this interview judge for yourself how good his english is...