A motorway is very strictly defined and will always be named Mn or An(M). There are A roads sometimes called expressways that look a lot like motorways but they don't have hard shoulders and their signs have a green, not blue, background. They may however have signs forbidding pedestrians. This might be an issue in places like the Shrewsbury bypass (but you could take the old route through the town); in Milton Keynes there's a path running alongside, the other side of a fence - use it.
As someone who sometimes cycles on big dual carriageways (there are places where there isn't a realistic alternative, such as the A40 near Monmouth) I'm not sure it's a pleasant or smooth idea. You'll be likely to end up running on the grass verge in places and those are right then they look.
Single carriageway A roads can be worse for cycling on than dual carriageways but are probably better for walking or running. Once you're in north Wales, there's actually pavement alongside the A5 for much of its length (I've walked a little bit of it near Cerrigydrudion and I think it runs from there to Betws-y-Coed at least)
Remember to walk/run against the flow of traffic so you can see it coming and step to the side if necessary.