Any theme parks near Musselburgh?

No theme parks. The best I can think of is Jump In Trampoline Arena or tubing at Midlothian Snowsports centre www.midlothian.gov.uk/directory/27/skiing_and_snowboarding/category/160

Definitely no 'theme park'. The only place along that description in the whole of Scotland is M and D's over in Lanarkshire area. Most towns in Scotland are served by travelling shows, ie fairgrounds. Some of the seaside towns have more permanent fixtures but I'm not sure whether any in East Lothian do.

Would East Links Family Park work though? It's got a lot of attractions and should be accessible to you on the train from Musselburgh to Dunbar.

When are you there? There's a fairly big travelling fun fair in Burntisland just now and for about the next month or so. If you can get public transport to Waverley station you'll get a direct train to Burntisland. There's also a pool with waves and flumes there if you wanted to make a day of it.

I'd recommend East Links too but it is 3 miles from the train station! The bus from Musselburgh goes past the (0.5 mile walkable) drive to East Links and is a much better option. East Links is also pretty pricey - £15 each.

Yes if you can make the journey to Burntisland the shows are there, there's also a sandy beach with lifeguards, large play park/splash park and a swimming pool with waves/flumes. I was there yesterday You would go Musselburgh - Waverley then Waverley - Burntisland by train.