month-long theater-focused stay in London

While not a month this is a theater tour from my local Chicago area Paramount theater that might be on interest Paramountaurora.com

London Walks has done theatre themed walks, I’ve done a couple.

I did a quick glance for this week and don’t see any but you might plug in your dates and see if anything is of interest. I’ve done a Noel Coward walk as well as a general West End walk which was theatre heavy

Since I did (and loved) a theater focused week in NYC with RoadScholar, your question made me wonder if they offered one in London.
This probably won't suit, but it is interesting!
https://www.roadscholar.org/find-an-adventure/11996/Best-of-London-Theatre-on-Queen-Mary-2

Thank for all the good replies!

If you end up wanting to book a show or two on your own, the Todaytix app gives you
all the info you need and is an easy interface to purchase tickets, including rush tickets,
as well as entering any lotteries (Hamilton, etc.).

https://www.londontheatretours.com/
https://lwtheatres.co.uk/whats-on/theatre-royal-drury-lane-tours/
https://www.nationaltheatre.org.uk/your-visit/things-to-do/theatre-tours/

Took the Globe Theatre tour ages ago. Liked it.

EDIT: If seeing a production don’t forget to look here for same day tix as well: https://officiallondontheatre.com/tkts/

The Royal Albert Hall runs tours too.

www.royalalberthall.com/tickets/tours-and-exhibitions/royal-albert-hall-tour
www.royalalberthall.com/tickets/tours-and-exhibitions/behind-the-scenes-tour

I can't personally recommend this outfit but it looks potentially interesting: https://www.tourlondontheatre.com/tours

Rogue theater company in Ashland Oregon is doing a one week theater tour with 7 plays and accommodations etc.
They are a really great theater company in our area.
roguetheatercompany.com is the website.

Richard II and My Master Builder coming up look like they will be very good.