Uniform reducibility of representability problems for algebraic structures

Given a countable algebraic structure $\mathfrak B$ with no degree we find sufficient conditions for the existence of a countable structure $\mathfrak A$ with the following properties: (1) for every isomorphic copy of $\mathfrak A$ there is an isomorphic copy of $\mathfrak A$ Turing reducible to the former; (2) there is no uniform effective procedure for generating a copy of $\mathfrak A$ given a copy of $\mathfrak B$ even having been enriched with an arbitrary finite tuple of constants.