Short answer: The Quay, the canal towpath and Riverside Valley Park are flat, level and continuous — the canal path runs for miles without a gradient. The city centre is the problem, not the riverside: Exeter is built on a hill and the walk up from St Davids station is a real climb.
Everything beside the water in Exeter is level. The Quay is flat and pushchair friendly, Riverside Valley Park continues straight on from it along the river, and the canal towpath carries on south without a gradient worth the name. You could spend an entire day down there and never meet a slope.
Exeter's centre sits above the river, and the climb from St Davids station or up from the Quay into town is genuine. If you are pushing or wheeling, plan to go down to the water and back along the flat rather than looping up through the centre. Exeter Central station puts you at the top without the climb, which is the simplest fix.
The Guildhall indoor family garden is flat, free and warm, with baby change on site. The Custom House Visitor Centre at the quay is free and on the level. Both work when the weather makes the riverside less appealing.
Belmont Sensory Garden is the oldest public sensory garden in the country and is designed to be used by people with a range of needs, with a play park alongside it. It is away from the centre, which is also why it is calm.
The Underground Passages are not accessible — they are narrow, there is an optional crawling section and groups are capped at fifteen. Parts of the cathedral and the older streets around it involve steps and uneven stone. None of that is a reason to skip Exeter; it is a reason to plan the day around the river rather than the hill.
The Quay car parks put you directly onto the flat section. Park and ride is cheaper but drops you in the centre, which means coming down the hill and going back up it at the end of the day.
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