Is there a block structure on the U matrix of the SVD of a block matrix?

Question

Consider a matrix $A_{m_a\times d}=U_a D_a V_a^T $ and $ B_{m_b\times d}=U_b D_b V_b^T $ with $$AB=\left[ \begin{array}{cc} A &|& 0 \\ \hline 0 &|& B \end{array}\right] $$ Now when you do the $AB=UDV^T$, does the $U$ matrix have a block structure as well. What I see when I compute this for example with a random $A_{m_a\times2}$ and $B_{m_b\times2}$, the $U$ matrix has $U_a$ and $U_b$ contained within it but the $U$ matrix is not in a block diagonal form.For example $$AB=\left[ \begin{array}{cc} U_a &|& | \\ \hline | &U_b& | \end{array}\right] $$ So is there a predictable structure on $U$?