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I would say that a reasonable proportion of the electorate may have moved position
I don't think there's much evidence for this. E.g.
But yes a cursory Google hasn't found anything super up-to-date.
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I define "reasonable" as "enough to change the referendum result in favour of remain if the same referendum question were run today".
Of course, this is just my opinion. It can only be confirmed or refuted if there were to be another referendum run with the same question as before (which is unlikely).
Thinking of "the electorate" as a single entity like that is kind of channelling the "will of the people" argument. I would say that a reasonable proportion of the electorate may have moved position (along with a considerable number of young people now able to vote).
The major parties (which seem to want Brexit) don't want another referendum as they know the answer will probably have shifted to the other side. If it was, say, 52/48 in favour of Remain they'd be fucked to know what to do without alienating large percentages of their support either way.
As for the local elections, I'm looking forward to the breakdown of the "OTHER" parties, but mostly interested in the turnout data. I'm guessing it'll be way down:-
Nothing on the BritainElects twitter feed about remain/leave polls since Apr 4th.