the pertinent human factor question is: why did those decisions at the time look to the pilot to be the most appropriate responses given the information held and the options realistically available?
An excellent question, to which the answer must be that wetware tends to make bad decisions under pressure.
If Pete had been making the decision about going to Battersea on the ground he would have taken a lot more factors into account. He might have thought, the client is in Elstree and the weather is clear back at Redhill and towards our ultimate destination in the North of England. It will take the client 90mins or so to get to Battersea from Elstree by car at this time of the morning. In 90 mins by car from Elstree he could be at Redhill or Luton or Cranfield, the latter two of which have precision instrument procedures. Even if he still wants to go to Battersea, flying to Redhill and waiting is cheaper than parking at Battersea. After discussion on the phone about which of these options suits the client best, he could have finished his tea, had a bacon sandwich, and leisurely initiated his new plan.
So the trick is to give yourself time and options, neither of which are available in the air in highly congested airspace in bad weather.