The TSR2 tables (Dec 2017) show that of 31 daily TPE services at Newcastle, 27 stop at Darlington and Durham, and 12 at Chester le Street.
So that looks like 2tph at DAR/DRH except for a few non-stops, and roughly 1tph at CLS.
Northallerton is a bit harder to work out, with the Middlesbrough services to consider.
Hourly at Chester-le-Street? Manchester services at 08:30, 10:30, 12:30, 14:30, 16:30, 18:30, 19:30, 21:30 and 23:30 would meet the Monday to Friday requirements shown.
Southbound there's 24 shown at Durham and 25 at Darlington.
Durham's requirements could be met by Manchester services at 07:15, 08:15, 09:15, 10:15, 10:45, 11:15, 12:15, 12:45, 13:15, 14:15, 14:45, 15:15, 15:45, 16:15, 17:15, 17:45, 18:15, 19:15, 19:45, 20:15, 21:15, 22:15, 23:15, 00:15 so on average around 1.5 services per hour between 10:00 and 20:00 and 1 service per hour outside those times. Although, a later first service or an earlier last service could change that average to 1.5 services all hours.