I did this journey several times last year - mainly in the other direction. I was using an Inter-Rail pass, so the fares scenario is different.
The quickest way is London - Paris on Eurostar, and then the night train to Rome. If you can get good advance fares, this is probably also the cheapest.
The routing I took was normally Rome - Milan - Basel - Cologne - Brussels - London. I could get to Cologne in a day, stayed overnight, and went on to London the next day. Coming from the UK, you may be lucky in getting the 49 DB advance fare (London-Spezial), via Cologne, as far as Basel Bad. You'll then need (to get to Basel SBB and) a ticket for the ex-Cisalpino train to Milan, and onward on the Frecciarossa to Rome. Great way to see the Alps, and experience Italian high-speed, but you'll need to overnight somewhere.
Alternatively, you could try to get the 49 advance fare from London - Cologne - Munich, and an advance fare on the Munich - Rome CNL sleeper, both bookable on the DB website.
If you've got time, remember you can build in two stopovers of up to 48 hours each on DB advance fares (including those ex-London).
Just checked on DB for one random date in September (7th) and the 49 fare is available from London to Munich or Basel Bad. [Can't check fares more than 92 days in advance].