How to plan a UK train route avoiding the tube?

The split-ticketing booking engine TrainSplit appears to be able to do this. In the advanced options, untick all of the modes of transport you don't wish to use (untick everything besides "Train"). In this case it suggests travelling via South West Trains and changing at Clapham Junction to get an Overground train to Stratford where you can pick up a Norwich train, or changing at Reading, Richmond and Stratford for a similar journey.

It's also worth pointing out that the tube isn't always going to be as bad as on the old deep-level lines through the centre of London. I suspect that you'd have an easy time (in terms of step-free access at least) taking the Hammersmith & City Line from Paddington to West Ham and changing there onto the Jubilee Line to Stratford, and picking up your train there. This is likely to be quicker than going via Richmond. I'm not aware of a national-level journey planner than understands the intricacies of step-free tube access, on the other hand.

Similarly, just because it's a mainline station doesn't mean it'll have easy step-free access between platforms - as pointed out in the comments, Clapham Junction for instance has a few very curved platforms with large gaps which are a little tricky for people with heavy luggage. Other stations might not have lifts or ramps, though this is admittedly less likely to be the case for major interchange stations.

If the issue is accessibility rather than the tube per se then the TFL website allows you to plan your journey within London and specify various access options. Click on edit preferences and you are offered the choice between, Use escalator not stairs, Use stairs not escalator, Step-free to platform, Full step-free access.

A further option (less good than AakashM's and Muzer's answers but exactly what I asked in the title and included for completeness) is GWR.com has an accessible booking tool which specfically has an option to avoid the tube (the tool is meant to be accessible, but it has some extra accessibility options for the actual journey). Other train oeprating companies may offer something similar.

Another option to consider: Use the central London connections, but take a taxi rather than tube. There is a taxi rank at each London mainline station. It costs more, and can be slower than tube, but is the most convenient option if traveling with suitcases.

For planning only you could try Traveline SE (There are other Traveline regions) this allows you to select or deselect various modes of travel including Underground but does not provide any booking facility.

Traveline also allows you to choose journeys which have step free access and also if you need staff assistance.

It also provides apps for mobile phones

For Bristol to Norwich it provides a route Bristol/Paddington/Kings Cross/Cambridge/Norwich using a bus from Paddington to Kings cross (and in that direction I don't think you have to cross roads.)

and if you ask it to avoid central London offers several including Bristol/Birmingham/Peterborough/Norwich and Bristol/Chesterfield/Norwich

Occasionally the computer has a problem auto-routing. These work reliably.

1. Phone up the train company, or, a travel agent. They can access the system in ways you cannot via a website; for instance they can build an itinerary as a sequence of trains -- rather than only "connect the endpoints" as the websites do.

2. Learn enough about the train system to figure out a route you like. Then book each segment of the route individually (online still, just several 1-train rides which connect back-to-back).

If you are looking for an online system that will do all that for you, #1 that's a "shop-for-me" question, not allowed on StackExchange. #2 It seems like this would be a simple computing problem, but it's actually quite complex. Occasionally an auto-routing engine chokes on a route because of a bug. Fixing that would require a disproportionate amount of software engineering, so it will probably not happen. More complex routes and special requests make this more likely.

Upside, different engines will have different blind spots. Downside, there aren't very many - they're expensive to develop, so most consumer-facing sites use someone else's engine. If trying four websites doesn't do it, trying twenty probably won't either. There's no substitute for review by a knowledgeable human.