The Timetable World website has a collection of historic timetables of varying types and vintages available online at
https://timetableworld.com/timetables.php.
Some old timetables are available for free, access to other requires payment for download. You're in luck that the website offers the 1962 London Midland Region timetable for free:-
https://timetableworld.com/ttw-viewer.php?token=61ac8ba5-0d78-48c9-83d2-1c716097f90d (
not sure if this link works on another computer)
Table 50 is the one you need.
Timetables for long distance services from that era can be confusing regarding what were actually through trains and which destinations required a connecting train. Interpreting them is a bit of an art - the note 'TC' in the train's column is the key to start and/or end points.
A quick look at the relevant table seems to show only one direct train to Liverpool Lime Street which stopped at Nuneaton Trent Valley - the 08:30 departure from London Euston, calling at Nuneaton Trent Valley at 10:27 and arriving in Liverpool Lime Street at 12:38.
Unfortunately, much of the route between London and Liverpool was under major re-construction and electrification during 1962. The number of train services was restricted; some were diverted to other routes (e.g. you will find few London - Manchester expresses in this particular timetable - they had been diverted to an alternative route until the electrification project was completed). After 1968, there may have been more than one train per day between Nuneaton and Liverpool, but that probably does not help with your novel!
From Birmingham to Liverpool was a better situation. In 1962 there were through trains at:
- 08:15 Birmingham New St to Liverpool Lime Street (arr. 10:32)
- 09:20 Birmingham New St to Liverpool Lime Street (arr. 11:56)
- 13:50 Birmingham New St to Liverpool Lime Street (arr. 16:30)
- 15:35 Birmingham New St to Liverpool Lime Street (arr. 18:01)
Although these were all express trains, they did not appear to carry names; I don't know the cost of a ticket.