Is roaming free in the BVI with my UK (O2, EE, Vodafone) or EU SIM card?

No! This is the #1 roaming trap. Although it is a British Overseas Territory, the BVI is NOT part of the UK and NOT in the European Union. Your "Roam Like at Home" (EU) and standard UK carrier plans DO NOT APPLY. Most will charge you extremely expensive international roaming fees (e.g., £6 per day or £2 per MB). A Cellesim eSIM is a one-time prepaid fee that is significantly cheaper.