We are like you from the UK, we originally booked the flights with Cunard via our very good independent TA, this was booked before the flights had been released by the airline.
When details of the Cunard flights were released our agent gave us the details and we were able to book direct with BA quite a lot cheaper, we even managed to get PE seats for less than the Cunard economy price, and the agent was able to cancel the Cunard flight.
We are sailing out of Vancouver for Alaska though, therefore our flights are London Gatwick to Vancouver.