You can easily do the two trips the same day. You could even camp overnight (or stay in a posh hotel) and do Wood Mine and perhaps the Hough Level the next day.
What you need to do , imho, is to arrange a private trip to West Mine for say 10am and then book the last trip into Engine Vein or book the first trip into EV and then book a trip into West for a noon-ish start. This will easily allow four hours in West Mine.
It is highly likely that your West Mine guide will be up at Alderley for the Open weekend anyway.
Steve Mills , in my experience, will be happy to guide you if he is fit and well.
Nigel Dibbin is the DCC webmaster and the fount of all knowledge on the AE mines.
I hope you enjoy the weekend.