Eurostar trains have a passholder quota (unique in Europe along with TGVs out of Brussels) and since they operate on extremely popular routes trains can sell out days or weeks in advance.

This is the case here: all trains from London to Paris are sold out on 2nd May.

On 1st May there would be this:

- 07:01 - 10:19 2nd class only

- 15:31 - 18:50 2nd class only

- 20:01 - 23:18 1st class only

Let me see if I can find something else.

EDIT: only train option on 2nd May:

- London - Lille 07:04 - 09:26 1st class only

From Lille it's very easy to reach Paris. You'd need a 1st class pass.

Otherwise there would be Flixbus, ferries (slow but interesting way!), flights or a full fare Eurostar ticket. The latter two options won't be cheap.

All other European trains are much easier but the Channel is a big bottleneck and Eurostar has a monopoly so...

2nd class option on 2nd May:

- Eurostar London St. Pancras - Bruxelles-Midi 06:16 - 09:13 30€ 2nd class

And then it's possible to reach Paris relatively easily. For example:

Option 1:

- spend a few hours in Brussels

- TGV Bruxelles-Midi - Marne-la-Vallée (Disneyland) 13:17 - 15:09 20€ reservation

- RER A into the center (not included)

Option 2 (hourly trains)

- IC Bruxelles-Midi - Tournai

- IC Tournai - Lille-Flandres

- TGV Lille-Flandres - Paris-Nord 10-20€

Option 3: fully reservation-free via Charleroi and Maubeuge. I've just done this today!

Book the Eurostar ASAP. Feel free to ask anything.