The itinerary includes 11h32 of transport and 4 steps, with a minimum budget of 71 euros.

Bordeaux - Paris
The step from Bordeaux to Paris lasts 2h06. You'll travel in a high speed train.

Paris - London
You probably already know this, but as of today, Eurostar is the only train company connecting France and Belgium to the United Kingdom via the Channel Tunnel.
As for fares, I recommend booking at least a month and a half in advance if you want to find a round-trip ticket for under €100.
Another important point: since the United Kingdom is not part of the Schengen area, passport and security checks can take quite a bit of time. It’s recommended to arrive at least an hour before departure.
The step from Paris to London lasts 2h21. You'll travel in a Eurostar.
On board, you'll have access to: Wi-Fi.

London - Holyhead
The step from London to Holyhead lasts 3h50. You'll travel in a long distance train.
On board, you'll have access to: Wi-Fi, Drinks and snacks, Bike.
You can find tickets from 27 euros if you book 2 months in advance.

Holyhead - Dublin
The step from Holyhead to Dublin lasts 3h15. You'll travel in a ferry.
On board, you'll have access to: Drinks and snacks, Bike.
You can find tickets from 44 euros if you book 2 months in advance.