There is a high level of similarity between the two dishes, as they share an origin in the same Chinese dish. To answer your question, let's look at the differences:
Bami Goreng is an Indonesian / Malayan dish where the flavor is influenced heavily by the presence of Terasi in the spicing mix. This type of shrimp paste has a very unique taste that sets Indonesian cuisine apart from all other Asian traditions I have tasted.
Bami Goreng in the Indonesian kitchen is made from thick flat noodles made from wheat flour such as the ones shown here.
As far as I know the Singapore variety is always made with thinner rice noodles. A very important distinction as the two types of noodles have very different flavor and texture.
Where the taste/spicing of the two is concerned, the Indonesian version is mostly savory from the use of the Terasi, sambal and ketjap. What is typically served as the Singapore version is seasoned with Curry Powder and should therefore have a more spicy/sharp character.
In all honesty, all these distinctions rely on the correct naming of the separate dishes in the UK. They should be quite different, but in practice the same dish might be sold under both names.