Putting aside the issue of planning permission for the new dwelling you wouldn't be able to remortgage her house unless your name is put on the deeds, if she is remaining on them as well which would be wise she is going to have to be party to the mortgage too. Bank would probably prefer to finance the new house but if you are short savings for their requirements for that then that's an issue too and obviously all the planning/legal stuff would have to be 100% before they would give a mortgage on the new one.
Is the existing site big enough to split in two properly for both dwellings? My uncle did this years ago but sold the original house to finance the building of the second one. You are looking to release equity on a house you don't own (which can't happen) or best case scenario it's transferred to you with a right of residence for your mother which again bank might not like or it's in joint names, none of them is a great plan unfortunately.