What is this “Councillors of State” provision - and could Edward replace Andrew?
Counsellors of State are those appointed by the monarch to carry out the duties of the monarch when temporarily unable to do so in person - usually when she has been overseas.
The 1937 Regency Act spells out who they are:
1. The spouse of the monarch - Philip
2. The first four adults in the line of succession:
a) for the heir apparent - Charles - over 18
b) for the next three - William, Harry and Andrew - over 21
Edward served from 1985 until 2005 - his 21st birthday until Harry's and Anne from 1971 until 2003 - from her 21st birthday until William's. Andrew has been serving since 1981 and Charles since 1966.
But they also MUST be resident in the UK. Harry isn't living there at the moment and so would either have to be brought back to the UK for the duration of need to serve with William or they would go the simple route of having William and Andrew do what is needed as a pair.
When The Queen does leave us then Beatrice will step up and become the fourth Counsellor of State and Camilla will also be added to the list as the spouse of the monarch but Philip would drop off. They may do as they did in 1953 and pass a new piece of legislation (it was the Regency Act 1953) and all the spouse of the previous monarch to continue to serve but at nearly 99 I doubt that Philip would want that.
When Andrew stepped down last year it was announced that he would continue to serve in this role - if called upon. I suspect that nobody envisioned a position were only two were actually available.
For Beatrice to step up the government would have to either declare that Harry no longer resides in the UK at all or that for some reason Andrew was ineligible (and as he hasn't been charged with an offence, let alone convicted it would be an awful precedent to take away someone's rights based on a verbal allegation and evidence of a poor choice of friends).