Since Henry's MAIN MO re ending his union with Catherine was due to his 'need' to sire an indisputably legit male heir to keep the Tudor dynasty from dying with him and Catherine was clearly past childbearing, IMO, it's a virtual certainty he'd have proceeded with the seeking the annulment even if Anne had died.
Even though having Anne at the ready to bear the vital heir sweetened the pot, I can't imagine he'd have abandoned his plan and idly just waited for Catherine to eventually die THEN court the first eager lady-in-waiting wanting a BIG promo. Let's not forget that once he'd 'had it' with Anne herself, he wasted no time seeking a replacement.