I've just done some research in the online archive of the RCTS 'Railway Observer'. This is available at
https://archive.rcts.org.uk/the-railway-observer-editions/ and can be accessed for free with the registration of an account. (You don't have to be a member of the RCTS - I'm not!)
Each monthly edition has the latest loco reallocations for each company with the relevant shed codes in use at the time. Prior to 1935 it would appear that the LMS used a separate number series for each division - for example there seems to be M, C and W for Midland, Central and Western. Generally only codes are quoted without the depot names. I can't find any full list of codes but as some examples of named depots I've seen there's M10 Leicester, M33 Carlisle, C2 Low Moor, C6 Wakefield, C16 Wigan, W1 Camden, W1W Watford, W21C Caernarvon and W26 Edge Hill. As far as I can see there's no codes given for the Northern Division, i.e. Scotland.
Going back further to earlier issues the Midland Division codes are just given as numbers with no 'M' prefix so were these inherited from the MR? Anyway, I found in issue no.19 dated Sept 1930 on p.140 some details of recent shed visits to ex-LTSR sheds and the code for Plaistow is quoted as '34'. (Shoeburyness is '35' and Tilbury is '36').
Hopefully that provides some useful info - I wish I could find a full list of codes but maybe there's one lurking in one of the issues?