lmm6, as you can see, there are a lot of questions here. I hope you are reading all and will come back in to answer us.
I'm sorry you're in this situation, especially since you have never had a close relationship w/ your mum. iDK if you're looking for advice, but if so, I agree w/ those who suggest a senior day care center or a carer who will take her out, etc. Maybe one of the carers could take her back and forth to the center?
Also, I agree w/ the idea of moving your computer to the bedroom. That way, Mum won't join you every time you use it.
Perhaps establish some specific times you do spend together? To watch a favorite show on the telly or whatever? Then maybe she'll respect the time you need for yourself. IOWs, maybe it's time to establish some boundaries.
I don't understand why she has two carers if you say she "doesn't need" them. I suspect you're not really facing what her needs are or don't really know. Personally, in your situation, I think I would relish the time she spends w/ the carers, as that means that's less time she's irritating you.
If she has "perked up" since leaving the nursing home, either that's b/c of the natural course of her condition or living w/ you is really helping her psychologically. If it's the second, then you've done something good. But you still need time for yourself and to be able to call your life your own. I hope you can work that out. Keep us posted! Hugs!