AS can sell one, sure. Whether they will is another question. Looking at a random date, AS appears to sell AS-marketed, AA-operated and EI-operated codeshares to LHR. AS does not appear to have their code on any BA-operated flights from my quick search, and AS won't sell anything but AS-marketed flights through alaskaair.com as paid tickets. If I force it, I can make SEA-YVR-LHR or SEA-YYZ-LHR itineraries on AS and BA metal show up on ITA Matrix (with a BA codeshare on the AS-operated SEA-YVR flight or native AS on the SEA-YYZ flight), but they're not competitively priced on the handful of days I looked at. Seems to me like there are many reasons to be excited about the AS SEA-YYZ flight, but connections to Europe aren't very high on the list.

(Air Canada and WestJet do both do a substantial business selling USA-Europe flights via their Canadian hubs. But why go out of your way to fly BA via Canada when there's a nonstop from SEA which is priced much more cheaply all the days I looked?)