Maximum bipartite matching when some nodes must be matched
Assume the maximum matching problem is $U-V$ matching, and $S\subset U$.


Find maximum $S-V$ matching. If the result equals $|S|$, then it is a feasible matching in original $U-V$ matching problem. Othewise, no matching meets the requirement.
Using the feasible matching got in previous step as basis, improve it by augmentation, thus computing maximum $U-V$ matching.


Since an augmenting path never turns a matched vertex into unmatched vertex, the maximum matching found by above algorithm meets the requirement. 

Using Hopcroft Karp algorithm, the complexity is $O(m\sqrt{n})$.

I don't know if the $O(n^{\omega})$ or $\tilde{O}(m^{10/7})$ algorithm works. But basically any maximum matching algorithm that based on augmentation works. And for those unit capacity network flow algorithms, it also works as long as it accepts an initial feasible flow and arguments along it, taking the lower capacity bound into account. (ie. never augment back through edges $s \to x (x\in S)$)