There is a lot to unpack in this thread.
Whether the house should have been sold to cover care is irrelevant, clearly it wasn't, for all we know grandad may have been on CHC and therefore the NHS picked up the bill.
Your mum cannot easily force you to leave the house, you are living there and a part owner, the only way she can get you out is with an order for sale. Whilst these are pretty simple in some cases, they are more tricky in others.
When you own property jointly a trust is automatically established in law, and a judge ruling on the order has to consider if the purpose of the trust still exists - if he was satisfied the grandfather left the house as a home, rather than something to sell, and with the grand daughter and family in residence, and the mother not, he may quite well not issue the order for sale.
In that case there then enters the value of your mothers share, which is not 50% if you are unwilling to sell your share (and are not being obliged to) - an RICS surveyor / valuer would then have to value your mothers share as to be sold with a co-owner who will not sell - generally it is acknowledged that in such circumstances half a house is valued at £0.
Of course you will fall out with your mother if you tread this path.
It was a mistake to start paying rent, up til then you had 50% legal ownership and 100% beneficial interest, which would be a stronger position.