No shame in timeshares, as long as you're not the sucka buying them!
(No offense to your mother, she's a dear, sweet lady.)
Backpacking isn't necessarily some punk/DIY ethos where you have to do everything the communal way. It's about making the most of your trips with the least resources, right? So if you have the chance to save a couple hundred bucks on lodging for a week and put it toward other stuff (read: booze), then why not?
My friend works for one of those timeshare exchange companies, and she hooks me up with places to stay all the time. I'd rather stay in a hostel where it's more social and fun, but some places are expensive, or you might just want a little more luxurious break in the middle of the trip. Hell, I even used a timeshare week during this year's TP meet-up for Coachella.
One thing you can do is book the week, and stay a night or two at a hostel as well. That way you can make friends there, but you'll always have a nice place to go back to when you feel the need. (Not to mention you can, uh, bring someone back to your own digs... ya dig?)
As far as selection strategy goes, find out which place has the most expensive/poorest selection of hostels, and use the timeshare there. From your list, London's the most expensive (and a lot of the timeshares on exchange there are niiiiiice) but Paris probably has the paltriest selection of good hostels.