Teaser 2825: Twin sets

From The Sunday Times, 13th November 2016 [link]

The twins Wunce and Repete each made a list of positive perfect squares. In Wunce’s list each of the digits 0 to 9 was used exactly once, whereas in Repete’s list each of the digits was used at least once.

Wunce commented that the sum of his squares equalled their year of birth, and Repete responded by saying that the sum of his squares was less than the square of their age.

What is the sum of Wunce’s squares, and what is the sum of Repete’s?

As stated there are multiple potential solutions to the puzzle. In the comments I give some additional conditions that allow a unique solution to be found (which is the same as the published solution).