We're making a couple of assumptions here. Firstly that the train left Liv St without tail lights and that no-one spotted the omission. It's certainly one possibility but not the only one.
The despatch staff at both Liv St and Tott Hale watch the trains clear of their platforms and will report anything that is amiss. I find it odd that a train can get as far as Waltham Cross without tail lights and not be reported, even at such an early hour. Consequently it could be the case that the lights had failed or someone had been into the back cab and may have knocked the switch into the "off" position by accident. Equally, the lack of tail lights could have been noticed and reported and perhaps it had been agreed that the train would be stopped at Cheshunt or, better still, Broxbourne for the problem to be dealt with.
The reason in the Rules for reporting trains without tail lights is an old one, though no less relevant even today. The lack of a tail light is one indicator that a train may have divided and that a portion of it might remain in the section. This is more important for lines signalled by Absolute Block than for lines with track circuits, as there is no other way to prove that a train has cleared the section but to watch it go past the box and visually check that it is complete. It also explains why it is important to report the lack of a tail light immediately, as it could indicate a major risk to other trains.
That said, a train division would not have happened without the driver knowing it in this case. Plus the Lea Valley route is all Track Circuit Block, so no danger existed. The worst that could have happened is that they weren't changed over in London and no-one noticed. But the Rules must still be followed.
O L Leigh