This thread seems to be another opportunity for people to reply to a question with whatever pops into their head without doing any research to find the true answer. It really undermines the value of having this section of the forum, and I can't understand why people continually want to do that.
From a legal prospective if that service is left in the system by error do Northern legally have to pay up for taxis even though it's not on their strike timetable?
There is NO legal obligation for any UK Railway Company to pay for a taxi to carry any passenger anywhere.
So let's not start any further discussion that presumes this obligation is correct.
A journey which is "in the system" (whatever that means for a rail traveller is unclear from your question) does not create an obligation to convey a person anywhere. Passenger Rail travel in the UK is based on the legal principles of a Contract. A Contract is formed when a passenger buys a ticket (and there are other less common 'implied contracts' when a passenger travels without a ticket but with explicit permission of some form or other). Once the Contract is formed, the Railway Company(s) is bound to convey the passenger to the destination on the Contract.
There is NOTHING in those contracts which requires that journey to be by train. And, there is nothing in that contract which requires any rail Company to "pay up for" any transport which their contracted passenger has chosen to find by way of another Contract with anyone else (such as a taxi driver). It is only for the Railway Company, and could only ever be for the Railway Company, to choose how to convey their contracted passenger.
Hope this helps.
As for extreme circumstances such as hazardous conditions, then it would be arguably correct for the Contract to be interpreted in the context of those conditions, and in such circumstances, the parties to the Contract may be permitted to relax certain terms if they had become impossible, hazardous or unreasonable to fulfill.