Probably the simplest and least expensive London to Taizé journey is to travel to Chalon-sur-Saône by coach using BlaBlaBus. (Usually you need to change in Paris).
From Chalon take the local bus to Taizé. See timetable on the page By train and local bus.

 
The most comfortable way to travel from London to Taizé is by Eurostar train from St Pancras. If the tickets are booked well in advance the price may not be too high, especially outside of the summer period. (Tip: Try booking tickets from London to Mâcon-Loché all in one go on the SNCF site (English): http://en.voyages-sncf.com/en/ )
The journey is:
1) Eurostar from St Pancras to Paris (arrive at “Gare du Nord”) ( 2¼ hours)
2) Cross Paris (by taxi or RER-metro line D) to the station “Gare de Lyon” ( 30 mins)
3) Paris (Gare de Lyon) to Mâcon-Loché (by TGV high-speed train) ( 1¾ hours)
4) Bus from Mâcon-Loché to Taizé (see timetable at the foot of the page By train and local bus).

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Notes:
 Paris-Bercy (where the OUIBUS from London arrives) is within easy walking distance of Paris-Gare de Lyon, so it is also possible to continue your journey by high-speed train (TGV): see trains page.

 There are other London-Paris bus routes, such as Megabus, Eurolines, Flixbus, etc. which can sometimes be cheaper. The drop-off points in Paris vary however, and Paris-Bercy is the most convenient for the onward journey to Taizé.

 At certain times of year Eurolines run direct coaches to Chalon-sur-Saône, from where a bus connects to Taizé. The arrival time in Chalon is usually not very convenient.