I'm going to watch a show at the Savoy Theatre in London and planned to do some shopping beforehand. Will I be allowed to take this in with me? The website states that the cloakroom will not accept rucksacks, or suitcases??
When I saw a show at the Savoy Theatre earlier this year I brought an empty shopping bag with me (when I exited it contained the programme for the show). I've done that in several theatres, and also entered with a shopping bag containing the programme from a show I saw earlier that day, or the ticket for the show I was going to see later that day. I know I've had other small things in a shopping bag entering theatres in London, I also know that I've been at the Savoy theatre before, but I don't remember if I had anything in my bag back then.
As at most theatres, the guards at the doors wanted to look into my (and everyone else's) bag.
As you can't leave your bag in the cloakroom, you will have to bring it into the auditorium with you (or leave it, where it will probably be removed), so don't bring a ton of shopping. I just put my bag on the floor beneath my seat and like that everything worked.
You are likely to be allowed to take into the auditorium a shopping bag that's small enough to fit under your seat (they'll probably want to search it first for security purposes). There may not be very much space under your seat, some London theatres are quite cramped.
If you need to store something bigger (e.g. rucksack or suitcase) then I suggest one of the following:
Left Luggage at Charing Cross station (a few minutes walk from the theatre). Book here: https://www.left-baggage.co.uk/en
"Stasher" luggage storage service in the shop Gift London, which is on The Strand and looks to be very near the theatre. Book here: https://stasher.com/luggage-storage/london/stashpoints/the-strand-gift-london-856252379185 There's also a similar service named BagBnB which has a location nearby (it doesn't give the exact location until you pay - it may actually be the exact same shop as for Stasher): https://bagbnb.com/luggage-storage/london/trafalgar-square/luggage-storage-charing-cross-station
The "left luggage" services at the station are well-known and reliable. I don't know anyone who's used the Stasher/BagBnB services, but I'd expect them to be ok.
Hope that helps. BTW traffic in Savoy Court drives on the right-hand side of the road, don't get run over. :)