LNER telephone customer service has just said my ticket would be valid on the GC 16:12 HX-KX service.... let's see if that works....
Your train is automatically valid on that service, as it is the next Grand Central service. You don't need to obtain any endorsement or permission for that to be the case.
In this case of this particular cancellation, as the anticipated delay in arrival by waiting for the 16:12 Halifax-Kings Cross service is less than 61 minutes (18:42 vs 17:56), you unfortunately don't have the right to be re-routed at the earliest opportunity. Therefore you can't force GC to arrange for you to be able to travel via Leeds on LNER.
You will, unfortunately, also not be eligible for any compensation under GC's delay compensation scheme (unless the 16:12 arrives at KGX 14+ minutes late), as this has a minimum delay threshold of 60 minutes before you are entitled to anything.
I think you may nevertheless have a case for a price reduction (i.e. partial refund) under the Consumer Rights Act, as the reason for your service having been cancelled appears to be an ongoing shortage of GC traincrew. This is likely to constitute a failure to provide the service with reasonable care and skill, and/or in line with the information incorporated into the contract by virtue of section 50 of the Act (i.e. you likely chose to buy a ticket for the 15:25 rather than the 16:12 because you wanted to arrive earlier, and so the arrival time influenced your purchasing decision).
It's difficult to say how much any such price reduction should be, and I imagine GC may try to refuse to pay anything on the incorrect basis that having a delay compensation scheme excludes any compensation for shorter delays. But it's worth a try.