In order to get from London to Rome you will need to use
Eurostar train from London to Paris, where you can catch overnight to Rome.
For
London-Paris connection any type of pass doesn't make sense, since standard supplement for passholders is huge there - 57 British pound, while regular tickets can be much cheaper, especially if you buy those in advance (
eurostar.com)
Paris-Rome: special discounted tickets available, but you will need to book those in advance. For October there are tickets starting from 55 euro (
www.tgv-europe.com)
Rome-Pisa: same story with discounted tickets. Purchase in advance gives you up to 30% discount, so you can ticket for
Eurostar train will cost as low 28 euro (trenitalia.com), for InterCity train - as low as 20 euro. Or you can use Regional train - fare for it is standard and doesn't depend on time of ticket purchase, it will cost you 18.05 euro
Pisa-Paris: you don't actually need to go all the way to Rome in order to catch Paris train, because it stops in Florence, which is only an hour away from Pisa (ticket 5.8 euro). Florence-Paris discounted tickets start from 45 euro, but again you need to purchase them quite long time in advance.
About timetable: All the timetables you need can be found in on the official websites quoted above. Just wan to mentions, that your key train is overnight Paris-Rome, since it is the only direct train between Rome and Paris. You will use it twice - on the way from London to Rome and from Pisa to Paris, thus you need to plan you itinerary according to that overnight train. Means leave London in the afternoon, so you can catch it in Paris and leave Pisa in early evening, so you can catch it in Florence. Train connection Rome to Pisa and Pisa to Florence is really frequent, so no problems there.
If you have any specific preferences - we can reconsider timetable and choice of trains, but it will definitely require more changes on the way.
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