If you wanted to know the fastest route from Roma to Londinium in June of 300 AD or how much it would cost to ride the shortest distance at ox cart speed to Constantinopolis, check out ORBIS. Researchers at Stanford have created a model of Ancient Roman transport networks over land and sea composed of 751 points (cities, landmarks, mountain passes) and thousands of linkages.
Check it out at http://orbis.stanford.edu.
The fastest route from Roma to Londinium in June? A boat ride across the Mediterranean to Narbo, foot/army/pack animal across southern Gaul, and a coast-hugging boat ride from Burdigala will get you there in 26.6 days and 2,974 kilometers. That carriage to Constantinopolis would cost you about 2,087 denarii and would take 128 days at ox cart speed – perhaps you should consider a fast military march instead?