Most family holidays in the Czech Republic will begin and end in Prague, which is well served from the UK by low-cost and standard carriers, but you might also fly to Brno or to Vienna in Austria, from where a shuttle bus takes you the 110km to Brno.
By train, you can hop on an afternoon Eurostar from London to Brussels then change to a high-speed train to Cologne in Germany, from where there's a 'Phoenix' City Night Line sleeper train to Prague, arriving just after breakfast. Alternatively, there's an afternoon Eurostar from London to Paris, an overnight sleeper to Berlin, then a scenic EuroCity train to Prague. Or you can travel by daytime trains and make an overnight stop. For all details, see Man in Seat 61 (seat.61.com).