Eurostar is owned and headquartered in Europe so I have no doubt they are pressing the EU and French and Belgian Governments in particular (as SNCF owns 55% and Belgian state railways 5% of Eurostar) to ensure it continue to operate normally whether there is a deal or no deal.
If necessary the French immigration officials will revert to the continent and the UK ones back to St Pancras and immigration will take place on arrival rather than departure (as per airline transport)
Over a quarter of all UK-EU trade in goods goes via the tunnel so there is an imperative to sort this - just like there is with ferry traffic.
All of this brexit nonsense that is been spouted is just nonsense and will have little, if any impact on the OP who is now probably worrying unduly rather than planning their holiday.