A successful Franklin expedition and British Alaska

If Franklin succeeds, does Britain take Alaska from Russia and attach it to Canada?

Probably; an open northern route would give Britain reason to occupy its approaches and exploit a later conflict with Russia.

A successful Franklin expedition makes the northern route to the Pacific look strategically important rather than fatal and impractical. The British Empire had a strong habit of securing the approaches and stations along routes it valued. Alaska would therefore attract gradual commercial, naval, and political encroachment. A later Anglo-Russian conflict could turn that presence into formal annexation and incorporation into Canada.