I believe there will be a lot of confusion from non-trans people at best and more anti-trans hate and violence at worst. Some non-trans people feel their prejudices about trans people are supported by the government, however that is conceptualised.
There are hundreds, if not thousands of people like me who have little or no social history of trans treatment over decades. Depending on the pragmatic details of each individual, I suspect we will just ignore any attempts to stratify trans people out from non-trans people. On the other hand, I can also see some of us deciding to be political and deciding to purposefully create 'difficult' situations. I believe consideration of family, friends and social position will override everything though.
Others, with a social history of being trans, will be challeneged to act or react in the moment. It will be very difficult to live life as a 'non trans' person.
Lastly, non-trans women who are believed to be trans will find life more challenging. I know sex realists believe this never happens, but it does and they will be the recipients of the feelings of the same emboldened anti-trans individuals.
I know the confusion people experienced before I started using women's spaces. I know how young children reacted when their parents would initially respond 'I don't know' to questions of why the girl was going into the boys room and then ask subsequent, more difficult questions to answer without diving deep into gender and sex or waving off the question completely.
Maybe the end result of this will be a utopia where there is no gender and nobody cares about sex outside of reproduction. I'll probably be long dead by then.