The Deutsch Bahn website can give you a decent idea of the trains even though you have no need to go near Germany.
Trains if you set off at 0610 tomorrow:
Picadilly -> Euston
St Pancras -> PAris Nord - With Eurostar Ball aches
Gare de Lyon -> Torino Porta Susa - TGV probably not possible to carry a bike
Torino -> Parma
Parma -> Bolonga
Bolonga -> Rimini
The Bike carriage possible option only reallt works in germany, but they will tell you who operates each train so you can mash your head off the website of the companies.
You can also tell it what trains you don't want to use so can turn off HS by deselecting ICE..
But Eurostar is an HS Service, SouthEastern HS isn't.
You can also cover most TGV routes on a combination of TER, IC, ICN and EuroCity classed trains which makes bike carriage possible; around about this point it's worthwhile looking at InterRail passes...
A global pass lets you travel to and from port (international rail, sea or air) in country of residence; which is pretty handy!
Calais Ville dep 13:35 TER42615 Train Express Regional Direction: Boulogne Ville
2nd class only
Operator: SNCF
Boulogne Ville arr 14:11
Boulogne Ville dep 14:36 TER 2026 Train Express Regional Direction: Paris Nord
2nd class only
Operator: SNCF
Paris Nord arr 17:29
Paris Gare de Lyon dep 19:15 EN 201 EuroNight Direction: Venezia Santa Lucia
Sleeper and couchette train , Global price , Couchettes , Bordrestaurant
Operator: Thello
Milano Centrale arr 06:00
Milano Centrale dep 06:50 IC 583 Intercity Direction: Napoli Centrale
Global price , space for wheelchairs
Operator: Trenitalia
Bologna Centrale arr 09:14
Bologna Centrale dep 09:32 R 2891 Regionalzug Direction: Rimini
Number of bicycles conveyed limited , 2nd class only
Operator: Trenitalia
Rimini arr 11:05