In Uhlmann's theorem, should the polar decomposition be written as $A=|A|V$ or $A=V|A|$?

Ok I confused myself in the first version of this answer.

With the definition $|A|:=sqrt{A^dagger A}$, it should be $A=U|A|$. If you define instead $|A|:=sqrt{A A^dagger}$, then it is $A=|A|U$. Both definitions are common.

There is a left and a right polar decomposition $A=PU=UP'$ which are related to the SVD $A=VSigma W^dagger$ by $$ P=VSigma V^dagger, quad U = VW^dagger, quad P' = WSigma W^dagger. $$ Note that $Sigma$ is the diagonal matrix with the singular values of $A$, i.e. the eigenvalues of $|A|=sqrt{A^dagger A}$ (or $sqrt{A A^dagger}$, they are the same), on its diagonal. Clearly, we have $$ A^dagger A = P'U^dagger U P' = WSigma^2 W, quad AA^dagger = VSigma^2 V^dagger $$ hence $P'=sqrt{A^dagger A}=|A|$ and $P=sqrt{AA^dagger}$.