There are a number of different ways to reach Taizé from the UK by public transport:

Coach

Usually the least expensive London to Taizé journey is to travel to Chalon-sur-Saône by coach. This usually involves an overnight journey. There are various services: https://www.blablacar.co.uk/, https://global.flixbus.com/, https://www.eurolines.de/en/home/, etc. (You may need to change in Lille or Paris).
From Chalon take the local bus to Taizé. See timetable on the page By train and local bus, which can be checked for summer variations, etc. by looking at the LR701 service on this page: https://www.viamobigo.fr/en/mobigo-...

Train

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 TGV all in one go on the SNCF site (English): https://www.sncf-connect.com/en-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), or see timetable for the service LR701 at https://www.viamobigo.fr/en/mobigo-...

Coach, then train

It is possible to travel by coach from many points in the UK directly to Paris. From there one can continue by train. There are various coach drop-off points in Paris, however, and Paris-Bercy is the most convenient one for travelling onwards to Taizé by train.