It would be useful if TfL linked to the correct National Rail web page on their e-mail, but they don't.- https://www.nationalrail.co.uk/engineering-works/london-kings-cross-31-jan-20260131/
LNER are terminating at Peterborough with buses to Bedford for East Midlands Trains and Thameslink to St Pancras and central London
Lumo are not operating between Newcastle and London
Grand Central are not operating at all
Hull Trains are diverting into St Pancras
For local stations between Peterborough and Kings Cross/Moorgate a succession of rail replacement bus services are running as far south as Alexandra Palace.
From Alexandra Palace, Hornsey, Harringay, Finsbury Park, Drayton Park, Highbury and Islington, Essex Road and Old Street paper ticket holders from outside the contactless area can use TfL bus services.
Rail Replacement buses are also running from Stevenage & Hitchin to Luton Airport Parkway and Stevenage & Welwyn Garden City to St Albans City for EMR/Thameslink trains, also from Potters Bar to Cockfosters for the northern line tube.
For anyone with flexible (not Advance) tickets-
From York and Doncaster alternative all rail routes to St Pancras exist via Sheffield
From Newark (and Lincoln), Retford and Grantham alternative all rail routes to St Pancras exist via Nottingham
From Peterborough an alternative all rail route is via Cambridge into London Liverpool Street