Dr. Eamy -
You can do a direct route in the north in about 9 weeks but I would dissuade you from doing the Southern Tier in the summer. Yeah, I know you Aussies are tough old iguanas - but you'd enjoy the northerly route more - plus you'd have scenery that is quite different than that of Australia.
If you are content to start in the islands in Puget Sound and end in the islands in Chesapeake Bay - you can do it in two months. Plus doing the islands rather than the surf option means that it is far easier to get to the starting and ending points.
Friday Harbor > Kettle Falls > Glacier NP > Yellowstone NP > Black Hills > Big Sioux Riv > Mississippi Riv > Tippicanoe > Ohio River > Smith Island
9 weeks - 300 miles first week, 400 other weeks - that averages out to about five riding days of 75-80 miles (125-130 km) with two days off for hiking, exploring historic places, leeway for bad weather, etc. From experience, it's much better to plan a midweek start and finish so that you have fewer folks on the islands. Also more places open and better public transportation.
If you were to do something like this - - you could fly into Seattle and be out to the the San Juan Islands on an afternoon ferry. (You'd fly back from Washington, DC) You know it's really incredible that it only takes an hour to fly from Australia to Seattle!! Hee-hee. But you could decompress on the bus ride to the Anacortes Ferry Terminal and the ferry ride out to Friday Harbor. Plus there are bike shops at Friday Harbor since cycling is so popular on the islands. I've included a pic from the campground on the west side of San Juan overlooking the Straits of Juan de Fuca. It's a great place for a sunset.
Best - J