In reply to Tim Lairy:
The top boulders at Cratcliffe offer plenty of no handed options to test your balance on. For extra difficulty/stupidity, some can be done with no hands and just one foot or no hands, facing out from the rock. The Blob mantel no handed is good fun if you are flexible enough to get your foot up in the first place, do it on each foot and you'll have glutes of steel in no time. You could combine Cratcliffe with some of the lower boulders and Scoop Slab Traverse at Robin Hood's Stride. That should give you plenty to go at for a day or two.
There is also plenty to go at at the Roaches Lower Tier. The problems on the Pine Tree Slab and Blister Slab are all good for no handed climbing. I imagine there is also plenty that could be done no handed on the Upper Tier and the Skyline although I can't think of examples off the top of my head. If you ran out of things to attempt for a day at the Roaches, you could pop over to Newstones and do the slabs around Varicose.
At Stanage Plantation, you could do Pebble Flakes, the slab problems next to the Bullworker boulder and some of the problems on the lone boulder. I'm sure there must be plenty of other no handed options at Stanage too.
The problems on the Big Slab at Higgar Tor are good no handed (but quite high).
Cottage Rocks at Churnet has a good few highish slabs in the font 3-5 range that are good no handed but they feel quite airy if you top them out and I can't think of many other options to fill a full day at Churnet.
If you haven't done a lot of outdoor climbing, the above grit/sandstone slabs will probably teach you far more than a month or two of pulling on plastic.
Most trad routes below severe are probably doable without hands depending on your views on highballing/soloing. Or depending on the injury, you might be able to lead them only using your hands to place gear? Or you could get someone to set up a top rope if you want to make it safer.
During my recent layoff due to a shoulder injury, I also did a fair bit of one handed climbing which might be an option if only one shoulder is injured but if you do this, be really careful; I ended up having to resort to a lot of deadpoint moves which could lead to injury in the other shoulder if you aren't careful. I also spent a lot of time making my own no handed routes and problems on the slabs at the wall. Take this time as an opportunity to really improve your footwork and you'll soon be climbing harder than you were before.
Post edited at 11:08
