I had Harry who had to have his hind leg amputated after her sustained a compound fracture when he was about 6 months of age. The only help he needed in cleaning himself was with the ear on the same side as the missing hind limb. A hind leg amputee can be prone to gunky ears and if these are not carefully managed ear infections can occur. Overall Harry did very well for about 7 years. But he did develop arthritis in the hip of his remaining hind leg and he also developed arthritis in his spine. His pain was very well controlled for about a year on first just Metacam , then he needed Tramadol, Massage and Acupuncture too. As he became less active he also needed his bladder expressing every few days. Otherwise a lot of calcium sediment built up and that would lead to bladder sludge/bladder infections. We had to let him go when he was about 8 due to medication no longer controlling his arthritic pain.
I had another Bunny, Cerys, who was missing half of a front leg. Her Mother over-groomed her when Cerys was still in the nest. Cerys never had any problems with her limbs or spine and she lived to be about 8 too.