I know that advanced tickets are for the relevant booked trains only, hence why they are cheaper. Living in Gillingham, Kent. A friend of mine had a Gold Season Ticket to get to London, as she commuted there. She had to go and see someone, and needed to catch the earliest train from Kings Cross. In order to do that, she had to catch the first train which was the 04:09 to London Bridge. Unusually, the train was cancelled, and there was no way she would make her booked train from Kings Cross. After speaking to station staff, there was nothing they could do.
There's a lot of missing information, so I will have to fill in some of the blanks and make assumptions.
The 0409 is a booked connection with the 0615 East Coast service to Newcastle.
It is unclear if that train is what is meant by "the first train from Kings Cross", but assuming the intention was "the first booked connection of the day from Kings Cross, for a journey from Gillingham" then that's fine. If something else was meant, such as the 0550 East Coast service to Leeds, then we have a problem.
The next service from Gillingham is the 0434, arriving 0552. It is very possible that the 0615 would still be caught, depending on how well the Northern Line is running. However if the 0615 was missed, the passenger should be entitled to catch the next train, which would either be 0630 or 0700, depending on the destination (which we do not know).
Would she have been able to explain to the Train Conducter, or would she have been liable for Full Single fair, which she certainly did not have!!!
Providing she was booked on the 0615, then she would not be liable for any charge.
If, however, she was booked on the 0550 then that is not a valid connection, so it would be a matter for the conductor to decide whether or not to show discretion. I believe that approaching the conductor before departure would be the best way to maximise your chances, and asking nicely and being apologetic, explaining that the 0409 was cancelled and, crucially, showing the ticket for that portion of the journey. But the conductor would be entitled to charge the passenger for the cheapest available appropriate ticket (including a Railcard discount, if applicable) as it was not a valid connection. In practice, the guard may - again, at their discretion - excess the existing ticket up to the cheapest available ticket.