Is the new job in the NHS too?
I had someone do something similar recently and with HR advice, I had to let the new trust know that my reference was incorrect and the sickness was incorrect and had increased.
If you are actually stressed, and feel unable to work, fine.
If you are just wanting a few weeks off to recover before you start a new job, then use your annual leave or have unpaid leave or negotiate a new later start date.
I don't think it's particularly reasonable for the tax payer to pay you to go off sick so you are refreshed for a new job that you have chosen to go to, when there are other options.
And it will probably depend on your GP. IME, GPs are being a bit stricter about signing people off with stress at the moment - most people are stressed (work, home schooling, job worries etc) and more GPs seem reluctant to sign people off at the moment, but maybe that's just the area I'm in.