We have a SIM card that we bought in England and used in our unlocked phone. Would it work in Belgium and the Netherlands? If not, will one purchased in Brussels work in both countries?
Yes, so long as you haven't used it for so long that it has been cancelled.
SIM cards from one EU country work in all other EU countries, and usually also in non-EU countries in Europe. Your card will keep it's British phone number regardless of which country you are in. The charges will be slightly higher for "roaming" in another country, say + 50%, and you will be charged a small fee for receiving calls in a foreign country.
Always dial the full number, including country code, even if you are in the same country to make sure you get the number you want, and not a UK number.
Make sure you know how to add credit to the card, for example via the internet, as you are unlikely to find a shop outside the UK for the company that sold you the card.