We did the Vatican/Sistine chapel, I think it was Wednesday morning, when the pope does his address in St Peters Square. Anyway - we walked right in and it wasn't very crowded at all. That was in early June, so maybe not the same in July.

Maybe it's a good idea to travel from Dunham to Newcastle and take the (overnight) ferry to Ijmuiden (Amsterdam) from Amsterdam you can travel to Brugge.
From Brugge you can go direction Rhine (Cochem is a lovely town and close to Eltz Castle)and then continue your tour to Bacharch.
I think this will cost you less kilometers and travelling time...

Thank you marcBE! I hadn't thought of that!

You can always get into see the Sistine - it is just a matter of how many other people are there seeing it with you.

If the Sistine is a major point of interest for you, I would book an early-entry tour. Or an after-hours Friday tour. It will be spendy, but worth it.