Do you have any suggestions for a day out, meeting the following criteria?

• A day trip by rail from Western Wiltshire
• Enough at the destination to occupy about five hours
• Decent cafe (doesn't have to be Michelin...)
• Destination doesn't look post-apocalyptic, which includes places that have become car-dominated

 
Those restrictions seem to exclude Weymouth. 

Nailsea? (That includes Tyntesfield, a rambling National Trust property, just to the north).

 

That's Swindon off the list then!

 

These days, private cars dominate so many places, but there are many excelllent sections within towns and cities which aren't overrun ...

Places I have enjoyed by train from West Wiltshire include:
Bath
Bristol
Frome
Oxford
Salisbury
Stroud
Gloucester
Cardiff - Central and Bay
Barry Island
Penarth
Salisbury
Hamble
London
Marlow
Dawlish
Exmouth
Poole
Weymouth
Chepstow
Worcester
Malvern
Bournemouth
Southsea
Tiverton (by bus from Parkway and canal walk back)
Isle of Wight (if you allow a ferry and have a deep pocket!)

And places that I really should take a (further) look at might include:
Exeter
Reading
Cheltenham Spa
Chichester and onwards toward Brighton
Sherbourne, Gillingham, Axminster, Honiton
Didcot, for rail centre

Less that 5 hours (for me), but fun:
Yetminster
Avoncliff
Severn Beach
Lymington
Brockenhurst
Bicester Village
Oldfield Park (to Midford)
Yatton
Burnham (for Highbridge)

Overrun with cars IMHO from the station might include Dorchester, Weston-super-mare and Newport

Not listing post-apololyptic places, nor "home" places within West Wiltshire.  Many more options if you allow a relatively short bus ride from a railhead.  And others if you are looking for countryside / canal walk

 

Slight bias involved?  .  I have visited Nailsea, as friends live there. Would like to visit Tyntesfield again, but there doesn't seem to be a safe walking route between N&B station and the NT site?

 
Have you thought about Southampton or Windsor? If you don't mind two changes of train (at Reading and Richmond) and like gardens with a bit more, you might also consider Kew Gardens.

 

A number of years ago I did a walk from N&B to Bristol going through Tyntesfeld, Ashton Court and the Clifton Suspension Bridge.  Tyntesfield was accessed by a "back entrance" (stop giggling all of you at the back there) footpath from around the Wraxhall area.  It did involve a bit of roadside walking and a yomp though Nailsea but all in all I quite enjoyed the day out.

 

Probably rules out Slough too.

 
Exeter Central for shopping, museum, cathedral, riverside. It's not car friendly.