Abelian group $A$ is cotorsion if $\rm{Ext}(F, A) = 0$ for every flat $F$, or, equivalently, $\rm{Ext}(\Bbb Q, A) = 0$
Every $\varprojlim^1$ of an inverse system of abelian group is cotorsion, and, conversely, every cotorsion group is a $\varprojlim^1$. Proof can be found in Warfield, Huber. On the values of the functor $\varprojlim^1$.
If every group in the inverse system $G_i$ is f. g., then $\varprojlim^1$ is $\rm{Ext}(A, \Bbb Z)$ for $A$ torsion free countable: take $A$ equal to direct limit of $\rm{Hom}(G_i, \Bbb Z)$. In particular, that limit is always a divisible group.
Converse is also true, but version of proof I know is messy and uses structure teorem of alg. compact groups. Probably you can find something on that in Fuchs or Jensen books.