After the fall of the Western Roman Empire, Islamic scholars are credited with preserving Greek sciences. Did the Byzantines not carry on these traditions? If not, what was the state of medical practice in post-Roman Europe?

In the specific case of Early Medieval England, quite decent, actually. More can always be said on this matter, especially from anyone else with medical knowledge about other quarters of the Medieval world; for the meantime, OP, here's a few previous threads on Medieval medical matters:

u/sunagainstgold and u/Valkine examine what Medieval doctors were pretty good at treating, and sunagainstgold also looks at the study of medicine at the University of Pavia;
and why I pointed out Early Medieval England, u/BRIStoneman on a 9th Century text called Bald's Leechbook. This post covers the Leechbook's recommendations for infections, and this one shows a range of treatments from the book.

Bonus: Bald's Leechbook is available in all its digitised glory here.