Try DB London Spezial from London to Köln and an Europa-Spezial Tschechien for daytime travel to Köln (around 4-5 hours from London) and an overnight trip in a 6-berth couchette from there. You'll need to buy two tickets (in euros) from www.bahn.de . A cursory glance suggests that £150 (one way) is around the minimum you'll get this for if travelling in a few days' time. I'm sure it would be a great journey though - www.seat61.com will give you more information. Splitting at Dresden may also be worth investigating, but you'll probably need to enter either a 1 hour stop over in Bruxelles-Midi or 2 hour stop over in Köln Hbf to get the 15:04 (or similar) Eurostar + ICE + City Night Line itinerary for which through tickets will be available.
If you can get a good train connection and price, it's a great journey by train. However, Wizz Air have a flight for £35 from Luton to Prague on Tuesday morning and evening.
The other option is to look for DB London-Spezial tickets for London to Dresden, which would mean a daytime routing of Eurostar to Brussels, ICE to Frankfurt, followed by IC/ICE to Dresden. That would get you very close to the Czech border and two hours train ride away from Prague.
I've had a look on the DB website and there is a fare of 149 Euros from London to Dresden on Thursday, departing at 06.50 from St. P arriving 19.28 in Dresden Hbf, changing at Brussels, Frankfurt and Leipzig. Unfortunately, the 21.08 train from Dresden to Prague (which is a sleeper heading to Budapest) won't give me a price - it would give an arrival time of 23.26 in Prague hl.n. --- old post above --- --- new post below ---
As for the via Cologne, next Wednesday,
There's a 49 euros fare from London to Cologne on the 15:04 leaving St.P and arriving Cologne Hbf at 20:15, changing in Brussels from Eurostar to ICE. Then the CNL sleeper leaving at 22.28 would be 59 euros for a seat or 68 euros for a bed in a shared berth of 6, arrives Prague hl.n at 09.26 on Thursday.
All advance fares, so like the UK, they will almost inevitably increase as time ticks on...