I don’t think anyone here can answer the OP’s question reliably as there are so many technical variables involved.
It can boil down to who is funding the place in the home (i.e. self-funding, Local Authority or NHS etc) and whether it is actually a nursing home (where nursing is given) or a care home (where only perosnal care is given). Or even a hybrid if both types of care are needed.
Depending on the actual scenario from the above, the OP’s friend’s wife may or may not keep her relevant Mobility allowance.
The OP’s friend maybe better getting all the facts involved (if even known yet) and to approach a Welfare Rights organisation or CAB etc as it may affect other finances/benefits etc as well, particularly if Local Authority funded (or part funded) due to the financial means test (carried out seperate to the needs test).
A rough guide to the complexity is here, including the last sentence:
