Natural Transformations and Parametricity
Unfortunately, the remark of Wadler is too cryptic for me to tell what use he wanted to make of "lax natural transformations".  Here is a guess.  Relation-preservation squares can often be recast as lax commutative squares.  This is how they used to be written in old automata theory papers/books.  See paragraph 1.2 in my Notes on Semigroups.  To do this kind of thing, you have to mix up relations and morphisms and pretend that they are the same.  I am also not sure that it buys you anything new.  It is just uglier notation for saying the same thing as relation-preservation.

Please feel free to explore the connection, but I am not confident that you will find anything new by doing it.