Liverpool the A5058 Queen's Drive forms an almost complete semi-circle around the city, about 3-4 miles out from the centre. 1930s dual carriageway, now with some grade separated junctions, although it peters out at the southern end. I have only driven bits of this some time ago so can't comment on what it's actually like, although the map gives a fair idea.
There are some sections of another, uncompleted ring road another couple of miles further out called Woolton Road and King's Drive in the south and Princess Drive in the east.
Manchester the A6010 forms a half-circle around the east of the city from Chorlton-cum-Hardy to Cheetham Hill, about 2-3 miles out from the centre. Used by the famous No. 53 bus route, scene of Manchester's first bus-to-tram conversion in the late 1920s. Mostly suburban and inner-urban single carriageway with many signal-controlled junctions, but Alan Turing Way through East Manchester near the Commonwealth Games stadium is a high-quality modern dual carriageway (although ludicrously still with a 30 mph limit). Obviously this was designated as a ring road in the same way as the A4040 in Birmingham, although not completed (it seems to stop at the Manchester city boundary).
There's also an inner ring road which is variously the A57(M) Mancunian Way (with its notoriously hairy junctions), A635, A665 and A6042. The last link in the chain to the West near Granada Studios is currently being built. This is (I think) signed as a ring road.
Also, as I mentioned, two short, separate stretches of residential street in run-down areas of East Manchester called "Eastern Bypass".
Bolton - the 1930s A58 bypass forms a semi-circle around the north of the town. Arguably a bypass rather than a ring road, but it looks almost semi-circular on the map and is much further in distance than going through the centre. Runs past the historic Smithills Hall and Hall'i'th'Wood, where Hargreaves invented the Spinning Jenny. Includes in its western section what must be the widest two-lane single-carriageway road in Britain, and also a particularly bad example of dual-carriageway "one lane syndrome" (a perfectly good DC reduced to one lane each way by installing a cycle track and hatching, thus making it impossible to overtake even a milkfloat.)
There's a good picture of this kind of thing (albeit in Glasgow).
Peter Edwardson, 26th Mar 02