Hi Lawa, so sorry you're going through this.
There is a NKC trial that began in Liverpool that I think is now in Birmingham (Heartlands?), led by Professor Siobhan Quenby. There is some information on the Miscarriage Association website, or could try the hospital directly. This is, I think, the only trial where they do a sample of NKCs in the uterus rather than more general (and expensive) blood tests.
There is also a section about immunological testing on HFEA website the which gives the "official" information.
Best wishes.