No you do not have to get tested at A&E, you can say NO.

I was admitted to hospital in England and they tried to test me. I said NO and although they weren't happy, I was treated regardless.

It's not a legal requirement to test before being admitted for treatment.

The only scenario they can mandate testing is for surgical procedures, for the week I was in hospital I never tested once. But when I had to go for the camera into my stomach they would not do the procedure without a PCR. I probably could of took the matter further and disputed it, but it was only 1 test so I acquiesced (against my better judgement).