Interesting you find flying West worse. At least on 12ish hour flights (Asia or the US) I find the westward leg is just a really long day which, provided I can force myself to stay up til a sensible bedtime, I get over almost immediately. Eastbound, the super short night on a plane followed by a morning landing when you feel you should be in bed, screws me up terribly.
Your tip of switching to the new timezone as soon as the door closes is particularly helpful. It’s easy to fall into the trap of following the airline’s meal schedule and suffering for it later.
I find the gradual time zone adjustment difficult when travelling for work (too many early meetings and business dinners) but I’ll be more structured on my next vacation.
Where do you get melatonin in the UK, I didn’t think it was generally available?