> If Parliament refuses to carry out the people's wishes, would this be a case in which the Queen would be justified in stepping in?
No.
Parliament authorised Britain's EU accession with the 1972 European Communities Act. It then consulted the people in a non-binding referendum. It has every right, Constitutionally, to do as it pleases. Note that the "Crown only dissolves Parliament before a general election under conditions laid out in the Fixed Term Parliament Act 2011 (dissolution)" (emphasis mine) [1].
No.
Parliament authorised Britain's EU accession with the 1972 European Communities Act. It then consulted the people in a non-binding referendum. It has every right, Constitutionally, to do as it pleases. Note that the "Crown only dissolves Parliament before a general election under conditions laid out in the Fixed Term Parliament Act 2011 (dissolution)" (emphasis mine) [1].
[1] https://www.parliament.uk/about/how/role/relations-with-othe...