It depends on what registry the mare and stallion are registered or approved with. The most common registries in my area are ISR, Weser ems, and NASPR.
I prefer the ISR because they offer more inspections in my area. I have two welsh mares that I plan to eventually breed to GRP stallions. Those foals will be registered with the ISR and also as half Welsh.
At least till this point, Westphalian has not offered full services in the US. This means foals were not even eligible for premium status and they had even less support to breeders than the more established registries in the US. As a long time GRP breeder in the US, we need more services...not less!
National Show Horses are part Arab, part Saddlebred. A GRP/Arab cross could only be registered half Arab (and potentially one of the riding pony books, if the Arab parent was approved).
National Show Horse Registry rules have changed, any horse that can be registerd as a Half-Arabian can be registered as a National Show Horse. The other half does not have to be Saddlebred anymore (and when the other "half" had to be Saddlebred there could actually be 2nd generation crosses that were 75% Saddlebred 25% Arabian).