Hire a Professional mover: This is probably the best way to save time on the move itself. A professional mover relocated houses as well as office relocations for a living, so they know exactly how to pack the truck, are fast and efficient and will have 2 or 3 guys or gals to carry and load furniture and boxes. You can help by taking boxes out to the truck and safely stacking them (out of the way of course) or passing them up to the person packing the truck.
Be organised: Be packed and ready to go. If need be, get up a little earlier to make sure you have time for breakfast and then pack up what's left in the kitchen. Running around doing last minute packing is going to slow things down, especially if the movers are having to try and find you for any reason.
Make a list: Moving house can get pretty chaotic so it's best to be as organised as you possibly can and making a list of things to do and remember is going to help you and the movers out a lot. Setting reminders on your phone can help out too so you don't forget to organise something.
Have all you documents at hand: This goes for anything you may need. Lease agreements, moving quotes/contracts, insurance papers, paperwork for storage units etc
Disassemble furniture: It helps save a lot of time if beds and the likes have already been disassembled, even if it meant sleeping on the floor for the night. Don't forget about taking drawers out and mirrors taken off dressing tables if these are too heavy.
Make a floor plan: If you can plan out exactly where all your furniture is going to go in the lounge etc. Make sure you have clearly labelled boxes so movers know which room they are to go in. You can even label bedrooms doors BEDROOM 1 etc and also on the boxes so each boxes is delivered to the right bedroom and you're not having to search for boxes.
Get some help: Get a few mates to pop over and help out too. Most mates will be happy with fish'n'chips shouted after moving along with a few cold ones.