You don't need ITA here. CO doesn't charge fuel surcharges when redeeming miles for either CO metal or their partners. They will charge other taxes and fees that will come to between $100-$250 when flying to the UK from North America.

I would definitely recommend transferring to CO. While BA miles have their place and are good for certain things, they are not good for flying into London if you aim to minimize your out of pocket costs. You may find, however, that they may have slightly better premium cabin availability than CO and their partners do. This is only a slight advantage, however. You'll also want to factor in whether you have any exiting miles on either BA or CO, since 50,000 points won't be enough for a ticket on other carrier to London from NA.

If you aim for an economy ticket to London, you might just be better off using your Sapphire points to buy a ticket, since 50,000 Sapphire preferred points are good for $750 in airfare, and many flights are Midwest and East coast are at or under this amount. Plus you would earn miles for your flight.