Allowing someone other than the Sovereign or a regent to appoint the Prime Minister would be far more than a break with tradition. It would be a constitutional innovation of the highest magnitude and would raise serious issues about the legality of the new prime minister's appointment. It could even go to the Supreme Court. There is no statute, but the Cabinet Manual, which is generally considered definitive on these things, says in clause 3.2:
"The Prime Minister accepts office at a
private audience with the Sovereign, at
which time the appointment takes effect."
Indeed, a Sovereign who cannot fulfil this most basic duty is not fit to reign and a regency would clearly be needed if that were the case. (Which does not seem to be the case here.)
I wouldn't be surprised if the Queen asks for Charles to step in. After all, he opened Parliament. It could be done very tastefully from Clarence House.
Well, the original plan may have been to travel back to London for the appointment. But, then commonsense may have prevailed and it was decided that the more agile and fitter person travel to the Highlands.