Moving from one council/HA to another 200+ miles away

gwynlas

If your mother moves out then you are at risk of becoming homeless and as such might only be entitled to emergency accommodation  such as a hotel room either where you are or in FIls area.
If you are on tenancy then you could look for a swap.
Mother needs to ensure that  she is secure in mew partners home.

Ratkin007

As your mother is the tenant, if she were to move out she would lose the right to her home as it would not be her sole and principal home.  She should therefore terminate her tenancy and you would not have any right to remain. 
The alternative and better option is to have the tenancy assigned to you and then try to do a mutual exchange or be added to the tenancy as a joint tenant and then exchange.  Have a look at the tenancy agreement and see whether there is any information regarding assigning the tenancy.  It would not be a succession because succession only happens when the tenant dies. 

You could also try contacting housing associations to see if they take direct applications and how likely it would be that they could assist. It may be worth checking the councils in the area you want to move to because whilst unlikely, they might be able to assist and they would be in the best position to advise you.