Don't know of any bye law stopping you riding on Exmouth beach, saw to horses today enjoying the winter sunshine and water. Be aware that from the 1st April Dogs are not allowed on the beach if you bring your dog with you.
Yes you can, check the tide times before you go though. I think you can also ride on Putsborough beach in North Devon, and also Teignmouth beach. Ring the relevant council to check first though!
I knew someone who got bucked off into the water at Exmouth, her horse than set a course for Starcorss - the far side of the estuary! He did come back eventually but it took a while!
Oh, and make sure you post lots of pics afterwards!
I think you can check the actual rules on the EDDC website, but there was an article in the Exmouth herald a little while ago, and from what I can remember you are allowed on at any time in the winter (when the dogs are allowed on) and in the summer you are allowed on after 5 or 6pm.