It depends on the ticket you hold. If you have an Advance ticket then, with some exceptions which likely wouldn't apply here, you must travel on the services specified on your ticket. If your ticket shows either a GWR or Elizabeth line service then you would have to take that. If it doesn't, you can take any reasonable connecting service, though in theory you should follow the itinerary you chose when booking.
If you have an Off-Peak or Anytime ticket, you can take any train that complies with your ticket's route and/or operator restrictions. If you have a through Manchester to Maidenhead ticket, routed "via London", then this would be valid on any operator - GWR or Elizabeth line. It would also include the cross-London transfer from Euston (EUS) to Paddington (PAD), using the Tube and/or Elizabeth line.
My preferred route between EUS & PAD is to take the Northern line (Charing X branch) from EUS to Tottenham Court Road (TCR), and then the Elizabeth line to PAD. The interchange at TCR is pretty painless - to minimise walking, you want to be at the very front (southern end) of the Northern line train. Depending on whether there's an Elizabeth line service to Maidenhead/Reading due soon at TCR, you could stay on the train beyond PAD. Otherwise, again the change between Elizabeth line and GWR services at PAD is pretty short.