If the constant is allowed to depend upon the dimension, the estimate is simple. Let $A=O_AP_A,B=O_BP_B$, then $AB=O_AO_B(O_B^*P_AO_B)P_B$ and we are left with showing that if the product of two positive definite self-adjoint operators $X=O_B^*P_AO_B$ and $Y=P_B$ is $\delta$-close to a unitary operator $V$, then it is $C\delta$-close to $I$. Now, we do not really know much about $XY$ except that it is conjugate to $X^{1/2}YX^{1/2}$, so all eigenvalues are real positive. Thus, it will suffice to show that if a $\delta$-perturbation of a unitary operator $U$ has real positive eigenvalues, then $U$ is $C\delta$-close to the identity. However, if it were not the case, there would be an eigenvalue of $U$ that is $\frac 12 C\delta$ far from the positive semi-axis. If you now take the Gershgorin disks of radius $\delta$ and if $C>5n$, say (where $n$ is the matrix size), then there would be a connected component of the union of the Gershgorin disks that would not cross the positive semi-axis, so some eigenvalues would be confined there.
I'm still curious if we can get a dimension-independent bound, so don't hurry to accept this answer ;-)