You've got your work cut out there!
You're going to get a few opinions, so here's mine.
Pack of green Scotch Brite pads. Given the ingrained dirt, you might be better giving a soak with diluted TFR (traffic film remover) first. Test small area first. Wear gloves and only let it sit for a few minutes and agitate with soft bristle brush. Rinse off and treat tube sections a portion at a time with Rib Revive. Comes in 1 litre spray bottle and use in conjunction with a green pad, or stiff brush, then rinse thoroughly.
In terms of protecting, 3M make a marine conditioner for vinyl surfaces. At £30 for a 250ml tube it's not cheap. Apply with damp microfibre. Should help reduce UV damage, but regular cleans will keep on top of dirt becoming engrained. Rib Shop make something similar called Rib Conditioner.
The danger with covers is damp and mildew if you're considering that unless well ventilated which isn't practical