Pharmacists worked against that, last century
A Practice in Yorkshire used light pens to sign the prescription on screen, and have it print out, signed.
Pharmacists not at their (dispensing) Practice sought a ruling against this, that it didn't comply with the Medicines Act requirement to sign the prescription before dispensing.
Basically this was an effort to reduce the convenience and make life as difficult as possible for dispensing doctors.
Actually they prevailed and the practice stopped.
A while later a government consulted the Law Society on whether legislation was needed to substitute electronic signatures of whatever sort for written ones on paper wherever other laws said "signed".
The response they received may have surprised them - it was "no" and "signature in English law is the making of a mark, with the intent of signature, and "in writing": "If you can read it, it is in writing".
But GP prescribing systems are nowadays largely specified by Pharmacists, financial types, and people who think emailing Word documents is electronic management of data.
So don't look for that convenience to return really soon.