Let's look at it per TOC (Normal day service patterns used. Some (like Northern) may not be possible in 1 day.)
ATW: Cardiff Bay - Warrington Bank Quay will need 3.
AGA: Sudbury - Southminster will also need 3. (5:30 departure needs 2)
Chiltern: 1 change (Bicester Town / Islip excluded for works reasons)
EMT: Worksop to Grimsby or Spalding to Matlock need 2.
FGW: Hereford to Gunnislake needs 3.
FTP: 1 change (Or is it two due to the LIV-Newcastle service at Victoria?)
Southern: Tonbridge to Ashford International needs 2
Govia Thameslink: Still 2 if I look correct.
Londom Midland: Stourbridge town to Alsager needs 3.
London Overground: 3 as mentioned by OP
Northern Rail: Ellesmere Port to Bishop Auckland. (6??? Warrington Bank Quay, Liverpool Lime Street, Preston, Carlisle, Newcastle, Darlington) Area around Grimsby excluded.
Scotrail: Many options are 3 (Far north to Inverness to EDB to GLC/GLQ to anywhere)
Southeastern: St Pancras to Bromley North needs 3.
South West Trains: Camberley to Lymington Pier needs 3 (Brockenhurst, Woking / Clapham, Ascot / Ash Vale) (Island line excluded)
Not mentioned TOC's only need one change, often at their central location (Birmingham NS for XC)
As this was quite some work, and as I'm not even British, I do apologise for any possible errors, but with my knowledge, I hope it's close to being correct.