Is it possible?

I can see the logic - the same root stock and a branch of the variety of the same tree.  I'm no expert but I don't think the branch will be the problem (providing it is healthy) but the rootstock that has sent a sucker is perhaps not a good idea.  Suckers tend to be reversion to wild type (not always) and may not grow true to the original root stock.  If it was my tree, I'd be removing the sucker. 

If you want to keep the apple variety line going for sentimental reasons, perhaps you could buy fresh root stock from a supplier (tree nursery) and try grafting on to them, or maybe some aggressive remedial work on the original tree (you might need to get a tree specialist) to try and rescue it?

Cheers! Thanks for the replies, I intend to leave the "new tree" and remove the old one when the newbie is established, I'm afraid most of the trees are in a similar condition, it would be a shame to have them all  end up as fodder for the woodburner.
    I must admit I never thought of new rootstock, that's what's good about this forum. When we first moved here a cherry tree had blown over in high winds, I've just left it on the ground and now I have five cherry trees which all bare fruit all growing from the original tree, I do have a bit of a battle though trying to beat the birds to the fruit.