Short answer: The sea tractor crossing to Burgh Island is arguably better in weather — you are driving through the sea on stilts either way. The Pilchard Inn on the island is 14th century and warm. Venus Cafe in the lower car park has plenty of indoor seating.
This is the thing that makes Bigbury work in poor weather. When the tide covers the sandbar, a sea tractor on tall stilts drives across through the water carrying passengers over to Burgh Island. It costs around £2, under-tens travel free, and being in weather while it happens makes it better rather than worse.
Bring cash or pay by card at the pub on the island.
A 14th-century pub on Burgh Island itself, and the reason most people cross. Quirky interior, good beer, and thick old walls that do exactly what you want them to on a wet afternoon. It is priced for the location.
The art deco hotel is for guests only, so do not plan on looking round it.
At low tide you walk across the sand for nothing. The tide comes in from both sides and the two flows meet, cutting the causeway faster than people expect — so know when the last tractor runs if you are relying on it to get back.
Venus Cafe in the lower car park has sea views and plenty of indoor and outdoor seating, which is more shelter than most Devon beach cafés offer. The WaterFront at Challaborough, the next bay north, has a beach-front position next to the holiday park.
The main car park pays by app and the phone signal is poor — worse in weather, when you are less inclined to stand about waiting for it to load. Bring cash, or use the lower car park with its cash ticket machine.
Kingsbridge is twenty minutes inland with shops, the quay and the best eating in the South Hams. Plymouth is forty minutes with The Box — free entry, a mammoth and a 4D history film — and the National Marine Aquarium, whose ticket gives free re-entry for a year.
If you are wondering about the island itself, see whether Burgh Island is free to visit.
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