Best app for train ticketing

You can use any of the Train Operating Company’s apps. LNER, GWR etc. all sell each other’s tickets. Seatfrog is an upgrade app, you need to have a ticket first.

Looks like SeatFrog now allows ticketing

Not from what I can see, they are not issuing any tickets.
In the UK, unlike Europe, no third party will ever offer cheaper fares than booking direct with the train companies, whatever nonsense they may choose to state on their websites.
All are making their profit somehow out of you- some of them it's just more hidden than others.
Buy all your tickets from the same train company (irrelevant which one) and store all your tickets in their app.
Even with what Seatfrog purport to do- offering upgrades- you can do that in differing ways with all the companies, often on train. Unlike Seatfrog it is always at a fixed sum when purchased direct- not whatever the highest bid is.
Even in Europe the 3rd parties are only the cheapest in certain specific instances, usually cross border trains.
None of these 3rd parties are Charities- they are manipulating you and the ticketing system in the pursuit of private profit

What’s the best way to figure out? Which company runs trains for entire trip? We’re doing London to bath to Bristol to Liverpool in Glasgow.

London to Bath and later to Bristol is Great Western Railway. Bath to Bristol is no advance fare, buy on the day.

(The normal advice on this forum is that the easiest routr on arrival day at Heathrow is National Express Coach from Heathrow to Bath).

Bristol to Liverpool there are 2 basic ways (many other less likely ones)-
either Bristol to Birmingham on Cross Country trains, changing on to London North Western Railway at Birmingham. Normally you would book that as an advance fare Bristol Temple Meads (Bristol Parkway is out in the suburbs) to Birmingham New Street then advance fare Birmingham New Street to Liverpool Lime Street-
OR
Bristol back to London Paddington on Great Western, tube to Euston, Avanti to Liverpool Lime Street
(the first route is by far the simplest and usually when booked split as stated, cheapest).
Liverpool to Glasgow- you want to look for one of the through Transpennine Express journeys. Otherwise you are on a Northern Rail train to Wigan or Preston, where you change to Avanti to Glasgow. Whichever this is always cheapest as a through fare.

So there are multiple operators, but times and bookings will be identical on the websites of any of this multitude of operators.