Does a driver have the authority to make such decisions?
I do on my train as long as nobody finds out!
The trains I am on about are the 2Fxx (Woking stoppers) and they connect with the train from Chertsey which (if late) means the pax have a 29 minute wait for the up Basingstoke stopper (calling at Walton on Thames, Surbiton and Waterloo only) or the next up Woking stopper, there is no connection at Surbiton for the suburban stations.
Chertsey arrives xx07/ xx37
Up Woking leaves xx13/ xx43
Up Basing leaves xx06/ xx36
So my stoppers waits a minute or 2 every time it needs to.
The 2Fxx usually ends up catching the 2Oxx at New Malden or Raynes Park anyway and so runs under cautions to Wimbledon, by waiting that minute or 2 at Weybridge means I can run into Wimbledon on greens at line-speed and leave on time, no delay is recorded and no other trains effected but a lot of happy passengers that probably dont even realise we held the train for them in the first place and just think they were lucky.
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Pretty much the same way its done now with estimates of how many passengers missed connections and how late they would be added in, I reckon the figures would be shocking which is why it isnt done.
There must be computer systems with all the relevant information on to get the ball rolling, in my case above any of us going past Exeter would have been an hour late, but because the train was held everyone was no more than 10 minutes late (I wont get into the other connections en route because none were missed anyway), Reading was only 3 minutes late and Paddington was only 2 minutes late, not exactly the end of the world and much better than some of us being 60 minutes late.