To be sure the two plots aren't that different if you allow that The Avengers will naturally also have time-travelling weirdness in spades.
I was hoping that the two screenplays would have been written by the same person, so that any similarities in dialogue would be explained by coming out of his own head, but I see Escape Route was written by Michael Winder and Escape in Time by Philip Levene, being broadcast in 1967 and 1968 respectively.
The similarities begin around the 20 minute mark (in The Saint) when a man called Harry is introduced who seems to be some sort of fixer for the escape. He speaks with the same sort of stutter as Thyssen in The Avengers, who fulfills the same role. The nuts and bolts of the escape are actually arranged by a woman in both episodes. I realise the connection is very tenuous but there are similarities. I wonder if Levene watched The Saint, enjoyed the episode and subconsciously reproduced some aspects of it in The Avengers. If anyone has seen both episodes I would be interested to hear whether you catch these echoes or whether you think I'm imagining it!
Incidentally I wouldn't have noticed this myself but now I come to Google Escape in Time I find it is popular with foot fetishists because Mrs Peel is in the stocks with bare feet at one point. There is also a webpage about the suits worn in the episode. And to think people with special interests had to join clubs before the internet!
Another wonder of technology has come to me this week and I must mention it. I'm writing this on my phone at home but at work now have Dragon voice recognition software and can only say I am fanatically keen. My boss is so impressed she's thinking of getting it for everyone because of how efficient and fast it is. My main fear was it wouldn't cope with my flat Midlands accent (Birmingham council once installed a phone system which didn't recognize the local accent) but it's great. I suppose the only reason we use hands to work computers is they evolved from typewriters but using the voice is vastly superior.