UK trade with biggest partners, by year
The OECD has nearly the exact data you are looking for. I haven't figured out how to get it into a raw data dump. I am pretty confident that this problem has more to do with me than it does with the data available.
The WTO holds similar quality information. I think somewhere a UK govt body is publishing this. Not sure where exactly though.
The UN has everything you need but you need to sign up.


See this question for other trade related sources:

Can anyone recommend a resource for monthly trade data for Mexico from before 2010?