Never got a reply on my email, and coming back from hols didn't had the time and the mindset to chase it up, although on second thoughts, I should have done at least on social media.
None of the stations in Burnley appear to have a help point, which is surprising. There's probably a notice with a phone number to call for help. Not necessarily easy to see at 11PM though. In your situation I'm sure most train companies would have refunded what you paid for the taxi, but if doesn't surprise me if Northern didn't, they're not good at things like that. Threatening to go to the local paper with your story might have helped.
Burnley to Manchester Airport. I'm not aware of any help point. There are just the ticket machines.
Which station was it? Most stations have help points, where you can press a button and get connected to the train company. If you'd used that you would probably have got better results. Train drivers and police won't know the procedures in that situation, but the train company would have been able to organise a taxi.
The last train to the airport was about at 11:00pm and the station's desk closing at 10:00...the only people on the platform when my train cancelled was the train driver and police...when I asked them what I'm supposed to do in order not to miss my flight at 05:00 in the morning, the reply was "sorry mate, we cannot help you, better get a taxi"...