When the Prime Minister is temporarily incapacitated, we essentially have Cabinet government. In practice, few difficult constitutional issues are likely to arise from this at all, esp in current conditions (stable majority + crisis). https://twitter.com/sundersays/status/1247254289069858818?s=20

Key decisions likely to be taken by First Secretary of State, by chairing a key inner Cabinet subcommittee + Cobra meetings, legitimised by tacit consent of wider Cabinet & Commons for these arrangements. The specific powers of PM as an individual fairly limited (eg appointments)

If there was for some reason a consensus within Cabinet for a different "first among equals" lead caretaker than the First Secretary of State, that could happen. But it is unlikely (outside of a specific context, eg a coalition). Formal decision-making is largely collective.

In hypothetical situation of a long period of incapacity, the Palace would look to the First Secretary, the Cabinet and Cabinet Secretaru to propose/agree interim arrangements. Little reason to intervene if there is agreement in Cabinet & sustained (latent) confidence of Commons.

In hypothetical sustained incapacity (eg PM in coma), Monarch does have power to appoint a PM without advice from (incapacitated) post-holder. In practice, likely to look to Cabinet for advice (Cab Sec/HM private sec seek to broker consensus) to avoid political interventions.

The "Presidentialisation" of the UK political system is more about optics of power, politics and communication than it is about the constitutional position, where the PM is appointed to form and lead a collective Cabinet government, which continues to operate in their absence.

The role and powers of the Prime Minister
(Peter Hennessy submission to Select Committee inquiry, 2011; updating a 1947 Cabinet Office paper)
https://publications.parliament.uk/pa/cm201011/cmselect/cmpolcon/writev/842/m2.htm

Main grey area is appointments. If ministerial changes became necessary (made in name of Crown), a caretaker would be wise to proceed by consensus with a key inner cabinet sub-committee. I am not sure where/whether legal constraints on other appointments (eg Bishops, etc)

On powers of war and peace
- collective decisions, de facto as chair of key crisis Cabinet subcommittee & Cobra meetings.
- the incapacitated PM's written instructions (letters of last resort, to nuclear submarines) remain operable, and are not replaced by an interim caretaker.

Hypothetically, if incapacity led to resignation, 2 parallel sets of decision processes.
a) Cabinet could propose [interim] successor to Palace. (Maybe First Sec State [default]/somebody else)
b) Cons Party decides own leadership timetable. (Could delay, if so (a) a bigger deal)

We essentially have Cabinet government (constitutionally) when the Prime Minister is not incapacitated!

Prime Ministerial power (Presidentialisation) arises de facto from political power/control of Cabinet and Commons. No presidentialisation without that

Evening Standard editorial concurs: First Secretary of State role is as "first among equals" chair collective decision-making (without exercising the patronage powers) https://www.standard.co.uk/comment/comment/evening-standard-comment-dominic-raab-s-job-is-to-be-first-among-equals-a-spirited-effort-a4409131.html

Counter-argument from Robert Peston. I'm not persuaded by argument this gets more problematic in recess
* This is easier in recess (practically)
* No *new* constitutional issue in that
(if compare to recess, when PM is well)
* Confidence of HoC v.clear https://twitter.com/Peston/status/1247475537272061954