I would chance order of cities for faster coverage: London -> Germany (Leipzig & Berlin) -> Poland (Warsaw & Krakow)->Czech (Prague) -> Austria (Vienna) -> UK
In any case take Eurail pass covering UK,France, Germany and Austria.
In Poland and Czeck and Slovakia (passing through between Prague and Vienna) buy regular tickets at stations, much cheaper. Note that Bratislava Slovakia is only 15 miles from Vienna.
Commenting the cities I would skip Warsaw (there is not much to see besides getto, mainly old socialistic monuments/monumental buildings of the era) and replace it i.e. with Carlovy Vary in Czeck or northern mountains of Slovakia.
The more budget trip you do the more you spend time in Slovakia, Poland and Czeck in this order. My son used to travel there on 15 dollars/day including travel, accommodation (tent/hostel) and food.
Budjet tips:
1. Stay the hostels, you will find them there www.HIhostels.com, saves a bundle over hotels.
2. Visit and see historical centers, but do not drink or eat anything there, few blocks of busstops can halve the price.
3. Spend some nights on train traveling, get more covered (one night covers cross Europe East-West) an save on accommodation.
4. Buy food at supermarkets, do not eat in restaurants more than necessary, my son even used to have a cooker with him. A noodle lunch is around 1 dollar anywhere if you can make hot water.
5. If you want to eat or drink in a restaurant check the prices from menu and order from there so you know what to pay (even if you dont always exactly know what you ordered. Usually the cheapest eating out is pizzas and stuff.
6. Best budjet months to travel are June and September, the entire Europe is still warm, but prices (and flights from USA) much cheaper than July-August.