As you're away for 2 weeks, I'm guessing there will be a fair amount of luggage with you. In which case, I would not follow the recommendation of the footbridge at the London end (near platform 4 and the Gourmet Coffee outlet on that side of the station), unless you are confident in your own strength to carry a lot of/heavy luggage up the stairs.
My advice would be to use the lift if so, obviously if you are not taking a suitcase (maybe it's a decent size rucksack) then the footbridge would be fine. For quick access to the lift, I would aim for coach D or E on the 390. Then if you want to use the lift going down the other side, it is just beyond the ticket barriers on the right hand side. The lifts in Crewe are usually pretty quick (especially compared to the one in Shrewsbury, that thing while huge is very slow at just about everything) so that will not hinder you too much. The lifts in Crewe are a good size too, so should not cause any issue. The one for platforms 6-11 will put you between 6 and 11, next to the cycle rack and turn right for platform 6.
Obviously it depends on the times of your trains too, but most of the Crewe to London starters generally use platform 7, alongside platform 6. Using the same sort of connection with my cycle, assuming no big wait for the lifts, I can do that within 4 minutes and that is going from coach A at the back of the 390 to the north end of platform 7. If I was going to the far end of platform 7, to be nearest the barriers at most of Euston, it would take me no more than 5 minutes. Even my little sister, with a dodgy foot, could do the connection in 6 minutes!
So your 12 minute connection should not pose any issue whatsoever. Obviously, check screens when you get there (there's one opposite Pumpkin on platform 5 which usually updates quickly) and there's one next to the gents on platform 5, which takes far longer to cycle through the 3 screens of information.
If you choose to use the bridge at the London end, the one next to platform 4, you will be on platform 7 within 2 minutes. If it's an 11 car 390, aim for coach F or U I think, on a 9er I would go for coach G. You really can't miss the bridge, Crewe is really well signposted and there's normally plenty of people around to help.
Just as importantly, have an amazing time and I hope you will share the experience with us in the trip reports section. Especially things like the tunnel tour, I want to know more about that!
Glad to hear you had a good journey, I too like LNWR for my journeys up and down the WCML. Such a shame they aren't an option to Glasgow!