DEFINITIVE LEGAL ANSWER!!!!!
Its legal to take washed up seaweed off the beach. Its a problem of access. Technically, going on the foreshore is trespass, but you cant bathe in the sea (which you have a right to do) without crossing the foreshore, so its whats called a 'tolerance', ie something thats occurred since 'time immemorial' (defined as 1189 AD).
So you can WALK onto the foreshore and CARRY off as much WASHED UP LOOSE seaweed as you like. But if you want to take a car and trailer onto the beach and load it directly, you need permission from whatever Parish, District or County Council has jurisdiction. Some will not give it, some might make you buy a licence (and i bet none let you do it free).