I had never read the continuation of the Key to Time but apparently the writers have. I was searching the internet and came across 3 audio books that had been put to paper: The Judgement of Isskar, The Destroyer of Delights & The Chaos Pool. From these stories I gleaned more insight into the writers of Lost. Apparently they took elements of these stories along with the Key to Time and the Face of Evil took key plot lines shook them up and came up with Lost.
Excerpt from the Destroyer of Delights:
The White Guardian accuses the Doctor of being in league with the Black Guardian, but the Doctor insists he was left with little choice as time itself has started disappearing and the entire fourth dimension is collapsing to a point where it will soon be indistinguishable from the first. It occurs to the Doctor that the White Guardian’s powers must be waning too and he‘s stuck on this planet just like the rest of them. If he’s trapped in the form of the Legate, he’ll be experiencing the Universe through human eyes for the first time.
I’m not saying that everything matches 100 percent but you don’t have to be a genius to see that these events are similar and when put together with the mounting stack of other similarities that I have found you would have to be pretty gullible to believe that this is still a coincidence.
If you want to read the stories for yourself they can be found at the following locations:
The Judgement of Isskar:
The Destroyer of Delights:
The Chaos Pool: