I guess it depends exactly what was done by Fultons - if the 'action' only was bedded then changing the barrel would not have any effect. If the barrel is changed and they had done bedding to the forend (to suit the old barrel) then you would be destroying the bedding and would have to re-do it with the new barrel.
Whilst I personally have no experience of Fultons work I have been warned by two 'experts' to look closely at the rifles they sell and the work they do.
Allegedly they have been seen to have sold a No4 T as original when it had been 'made up' by them and the numbers changed to suit.