I am a non-permanent resident in the UK without a British or EU passport but have a nationality that in normal times grants visa free access for 90 days in the Schengen area. Looking at the French government information I can see the following:
I am a foreign national. Can I enter France?
There are no restrictions for travel to Metropolitan France from the European space (European Union Member States, United Kingdom, Andorra, Iceland, Liechtenstein, Monaco, Norway, San Marino, Switzerland and the Vatican).
Since the European space includes the United Kingdom, would this mean anyone who is in London legally and is eligible to visa free access in France can travel by Eurostar freely? Which would then allow travel to other Schengen countries.