You're right about the big sports brands having a bigger budget. The traditional brands may have a long term issue as NB, Adidas, Nike and Puma chuck money at the biggest stars. GM do seem to have fewer and fewer international players, Slaz have all but disappeared.
England's recent new players:
Roy, Willey - Adidas
Ballance, Wood, Stokes - NB
I would perhaps add Puma to the list of options for Root, but only because he uses their shoes. But, if NB are going to be England's kit suppliers, then this would be my guess. Otherwise, Adidas, who have a large swathe of young county players in their stable.