If you have an Anytime ticket, then you can use any train regardless of any reservations - counted place or otherwise.
If you have an Advance ticket you must travel on the reserved trains.
Does this rule definitely apply even to the non-reservable leg of the journey? If it does, then at least from my experience, ticket inspectors on local and regional trains, such as Northern and TransPennine, rarely ask to see your reservation if you show a ticket for a connecting journey on another operator's longer-distance train, so in practice you can often get away with using another suitable connecting train, particularly if it will give you a more comfortable connection than the one shown on your reservation.
I often travel from Sheffield to London King's Cross via Doncaster, rather than with East Midlands Trains via the Midland Main Line to St Pancras, as I find it's often easier to get cheap Advance tickets with Virgin East Coast (and occasionally with Hull Trains or Grand Central) than with EMT. If I buy a through Advance ticket from Sheffield to King's Cross, it often gives me a rather tight (say 10 minute) connection at Doncaster, and I generally prefer to take an earlier train from Sheffield to Doncaster in case the one shown on my ticket runs late, causing me to miss my connection.
Sometimes an Advance ticket says "On specified VTEC trains and connections" which could be interpreted to mean that you must travel on the booked connecting local service as well, and sometimes it says "Valid on the booked services below and required connecting services" which looks as if it means you may use any suitable train to connect with the main part of the journey.
Back in the 1990s and early 2000s, when Advance tickets (or Apex as they were called then) were often issued on large format ticket stock similar to that used by Eurostar, they used to show your booked train for the main part of the journey, and if there was also a shorter connecting journey, that bit used to say "suggested service", meaning you could use any other suitable trains for that leg of the journey - surely that was a more sensible solution.