Jura I don't think "illegal immigrants" is the correct phrase. We would be deemed third country nationals and subject to the same rules as Americans, Australians etc. In France, if we have been living here in a legal way, paying taxes, in the health system, then we would not be "illegal", but we would need to change certain things. The French Prime Minister has committed to helping us do that. For example, we don't know if the S1 will continue, but under the PUMA scheme we would have rights to health cover from the French state. Our titres de séjour might need a title change, but the conditions on which they had been granted would still apply. The French government is not being alarmist here, just re-iterating the rules which would apply when the UK is no longer part of the EU.
I think that workers are in a more difficult position, especially cross-border workers, as is anyone who has not lived here in a "regular fashion". I know a lot of people are very upset and stressed, but facts and actions are the way to progress, not panic.
I don't bother engaging in discussions about Brexit on here anymore because I see no point in continuing the same arguments with the same people, but I do think it important that we deal with it factually and calmly.
It is a terrible, unecessary mess, but I am far more worried about the disastrous impact on the UK and the future of my grandchildren than I am about us.