Parents home

Best to get a legal agreement drawn up so you’re both clear about what you think you’ve agreed. It wouldn’t cost much and could save a lot of hassle.

The biggest problem would be if she can’t pay the fees and applies for help. Then asset deprivation would be an issue as she could have income from rent to offset against fees. Also a charge could be put in the house.
It’s really best to get some qualified advice

As Peasblossom has posted, you should get some proper advice. Problems would arise if your mum's unable to pay the fees. She may have to start charging rent, have a charge put on her property or have to sell.

If she’s not able to pay her own nursing home fees, then I’m afraid it’s very likely that sooner or later the local authority will require the house to be sold in order to cover them.

I would just add that unless the house is particularly large or special, any rent will be highly unlikely to cover more than a small part of nursing home fees.

Though of course if she has a substantial income and/or enough savings anyway, the rent may be a useful extra.

I’m assuming that your Mother doesn’t have Dementia as she has told you that you can move into her home, it might be a good idea to get a POA drawn up so that if she needs anything done in the future you will have the authority to go ahead with it.
MIL went into a home and her POA had an error in it, it took ages to sort , luckily the Care Home waited for their payment for over 3 months as the bank wouldn’t release any funds to my SIL to pay for her care.
As the others say, get legal advice re moving into your Mums home, it might be ok in the short term, but you might find it’s a temporary solution if the house needs to be sold.

I would pay her rent-I'm sure you can agree something that all parties are happy with. Living rent free sounds a bit cheeky to me.

I agree about legal advice.
The main issue is as above, and also if you have siblings (or anyone else, who might have an 'interest'.
However, 'caretaking' in return for paying bills but not rent is not uncommon - I have done it. It can especially apply if for instance, you need to repair & decorate the property.
FYI 'short term' in regards to legal / residential is often regarded as 'up to 3 months'

I'd certainly look into both POAs if your mother has capability.

If the Nursing Home is going to be her future home then yes, you will probably have to sell the house at some future point to cover the fees. I had to do this some years ago for my mother.

You have to be very careful about Deprivation of Assets. Maybe worth checking with her local Social Services/Council/the Nursing Home re fees etc.