Nothing at all between Gloucester and the original Severn bridge.
You can cycle on either side of the original bridge, depending on which view you prefer. Although the entrance to the downriver path, on the English side is rather hidden up a lane.
I am surprised someone has said you can cycle across the new one, as there is no pedestrian or cycle path at all on it and the road is considered to be 'motorway' so I dont think it is possible to cycle via that one unless you have a death wish and long to get a fine.
Its sad they did not plan ahead better with the newer bridge – when the old one closes there will only be motor vehicle access to Wales. Particularly sad as crossing both bridges would have been a nice target for walkers and cyclists out on a trip anyway, never mind for access purposes when the older bridge is shut down, which it will be at some point as problems with the cables long term (rust).