The English anniversary is not a Simcha, so what’s the problem. Oh, you mean if you were married in Eretz Yisroel on Isru Chag and then moved to Chutz La’aretz. Aha.
There is a story about someone speaking to Rav Pam and he mentioned he was getting married on August 22nd and Rav Pam said that was his and his wife’s anniversary date.
Obviously, like most other questions of this sort, the correct answer may vary with whom is being asked. Based on her other postings, and assuming she had figured out how to pick the lock on the closet, Devny might have to consult her local Rav/posek as to whether its mutar to fast and say tehillim on her anniversary date if it coincides with a yom tov