Since we have numerous Barn Swallows on the place here in the western U.S. and since we're on the northern edge of the migratory patterns for the Swifts, I find the difference in wing shape to be telling.
All Swallows have an easily recognizable "shoulder" that's not so apparent on the Swift. The Swift's wings are a smoother crescent from the wing root to the tip while the Swallow's has the junction and sweep change just ouboard of the wing root.
Here in the U.S., the only Swallow that has the "streamers" on the deeply forked tail are the Barn Swallows. Not at all like the "sawn-off" stubbiness of the Chimney Swift.
Here's a nice link that can help, assuming similarities in the U.S. and U.K. species...
http://www.naturenorth.com/summer/Chimney_Swif t/Swallow_VS_Swifts-0409%20(2).pdf ...