It's just as illegal under U.S. law to have GCHQ do it for you as for NSA to do it directly.
This is a well-discussed loophole already and the courts have already made clear it's illegal.
E.g. you have a Fourth Amendment right to avoid having your postal mail (or any other papers or effects) searched unreasonably. You do not have a Fourth Amendment right to avoid having your UPS parcel searched by UPS.
Does this mean that the government can simply tell UPS to open your UPS parcel to evade the Fourth Amendment? It's not UPS's personal property so they have no Fourth Amendment protection to the parcel. And you have no Fourth Amendment protection from UPS.
But as it turns out once UPS acts under the direction and control of the government, they also have to comply with the strictures on the government. So they cannot legally be made to violate the Fourth Amendment on behalf of the government.
The big catch is that a third party may voluntarily divulge information (e.g. if they rip open your parcel by accident while throwing it out the truck on Christmas Eve and a bomb spills out, they can tell the government). But they can't do so on request of the government.
And so it is for GCHQ and NSA. GCHQ could conceivably tip off the NSA if they happen to get evidence of a US person plotting a terrorist act (much like the USA might tip off Germany to a German national, if it were discovered). But the NSA can't ask GCHQ to deliver information on a US person. At least, not legally, but if you're assuming the NSA will just break the law anyways you're already screwed.