It can be done in 6 days

 I really didn't think it would be possible, partly because of my slower walking these days and my desire not to leave hotels before 10am or home before 9am, and to end the day in time for an evening meal.

But I've found a way.

16 First Class TOCs in 6 days

In the above diagram: 

• Circles are changes of transport, rectangles are overnight stops.
• Those in green are the first time each one of the 16 Train Operating Companies (ToC) is used.
• Dotted lines are transfers between London or Glasgow terminals.

In case you've counted the symbols, I stay first and last nights at home and take two different ToCs trains from Kings Cross (KGX).

Day 1: Home to Victoria,
Gatwick Express to Brighton,
Southern to Havant,
Southwestern to Waterloo, transfer to Liverpool Street,
Greater Anglia to Ipswich and back, transfer to London Bridge and home. (10hr45)

Day 2: Home to London Bridge, transfer to Kings Cross,
Grand Central to York,
London North Eastern to Newcastle. Overnight in Newcastle. (8hr30)

Day 3:
Cross Country to Reading,
Great Western to Cardiff. Overnight in Cardiff. (6hr30)

Day 4:
Transport for Wales to Manchester,
TransPennine Express to Glasgow City, transfer to Glasgow Queen Street,
Scot Rail to Dundee. Overnight in Dundee. (10hr15)

Day 5: Scot Rail to Edinburgh,
Avanti West Coast to Euston, transfer to London Bridge and home. (9hr45)

Day 6: Home to London Bridge,
ThamesLink to Cambridge,
Great Northern to Kings Cross,
Hull Trains to Doncaster, Cross Country to Derby,
East Midlands to St. Pancras, ThamesLink to London Bridge, and home. (9hr45)

There are some long waits for connections, mainly because I need a train from a particular ToC. Longest is Doncaster's 2hrs, Edinburgh and Kings Cross are both 1hr30 and Manchester 1hr. There are plenty of things to do in and around these stations.

Coincidentally, and I've truly only just discovered this, all three of these stations have pubs within the station premises which are recommended in "Unusual Railway Pubs, Refreshment Rooms and Ale Trains" by Bob Barton.

Euston, Liverpool Street, Newcastle, Reading, St. Pancras and York are also mentioned for their bars. Not enough time at Reading and York but I may be able to have a sip in the others.