No, you won't. Don't worry. You would inherit from him (either under his will or if no will on intestacy rules - see latter below) and there is no stamp duty or inheritance tax I am pretty certain.
You can own a house either as both registered owners as 1. joint tenants or 2. tenants in common (English law). My father changes it from joint tenants to TC when he feared my mother might die second and leave her half not to the children but to a charity (although she would never have done that) so he wanted to be free to leave his half of the house to us the adult children - hence tenants in common. Both forms are having both in the couple on the deeds but it's something to think about if you do get put on the deeds.
If you plan to move in a few years' time it is probably best for now just to register your interest but not pay stamp duty and do a transfer and then when you move at that point get yourself put on the deeds or when the mortgage is paid off.
By the way if you are in England (not Scotland) and he dies without a will this is who inherits:
"The husband, wife or civil partner keeps all the assets (including property), up to £250,000, and all the personal possessions, whatever their value.
The remainder of the estate will be shared as follows:
the husband, wife or civil partner gets an absolute interest in half of the remainder
the other half is then divided equally between the surviving children
If a son or daughter (or other child where the deceased had a parental role) has already died, their children will inherit in their place."
So if your house is worth over £250k or may be worth that by the time you die wich could be in 40 years' time and you want it and not the children if he dies make sure you both do wills now leaving everything to each other.