I really like Overland Equipment bags. They are well-made, and some have water bottle pockets. None have wheels, but there are many sizes/styles. (I like Donner, but you may want slightly larger--though be mindful that gets heavy)
http://www.overlandequipment.com/products.htm
I feel safe with a cross-body strap.

We came from Gatwick in early March. It is gov't security rule not a rule put in place by the airline.

I had a large shopping bag and as long as all my things fit into that I was allowed to go thru security. Your purse, jacket etc all need to fit into one bag...no exceptions.

Once we got thru security I took out my purse and jacket and where never bothered again.