Ireland as a whole has never been a British colony. It has been invaded numerous times, dating back to an 1171 occupation, but during the time of the British empire it went from a kingdom (similar to modern Commonwealth countries, it was under the reigning monarch but not the country itself) to Northern Ireland, while the remainder of Ireland expanded to fill much of the island.
The situation of Bangladesh is complicated, to say the least. It was never officially a colony but did gain independence from the U.K. I will add Singapore, however.
Bangladesh may not have been a direct colony, but it was still a territory within the British Raj until it's dissolution in 1947. So shouldn't it still count?
Upon further investigation, Singapore went from a colony of Britain to being occupied by Japan to a semi-country state post-WWII to a merger with Malaysia and only then did it become a fully fledged country.
Did you leave out Hong Kong and Palestine because the British handed them over to other countries rather than granting them independence outright? If so, perhaps an additional explanatory note is in order, which would cover the Singapore case as well. Former colonies relinquished by Britian, which did not immediately become independent, sovereign nations are also excluded. Or something like that...
I'll add a note about this -- the quiz is just countries, not possessions which changed hands like Hong Kong or land claims like Palestine. (Although Palestine may have *some* claim to countryhood nowadays, though not much, it certainly didn't when it was released from British rule. It was then, officially speaking, part of the land area that was then declared Israel.)
British Cameroons? British Somaliland? Papua new guinea? Northern/Southern Cameroons were both separate colonies, British Somaliland was an official protectorate, and New Guinea was controlled by Australia. And the kingdom of Hanover was an English puppet until german unification in the late 1800s, but that wasn't significantly influential.
I think you should rephrase your second caveat that says 'colonies which were made part of Britain and never left', which I assume you mean the current British Overseas Territories. They were actually never a part of the United Kingdom and are not considered as such; they are dependent territories of the UK while not being a part of it.