Name change after marriage and BRP

Hello, I hold a Graduate visa. My partner and I will get married on 4 December. I will report the marriage to my national authorities and request a new national ID card and a new passport with my new surname. However, I am not sure what I need to do with my visa or BRP. If I need to replace my BRP, when should I do this? Before or after the passport change? Home Office website states that it should be done within 3 months but I am not sure whether my new passport would arrive within that timeframe. When would be the official date for the name change, the date of ceremony or passport issue date? Also, I still have not received my BRP since August 2021 for my new visa and pretty confused at this point. Thanks.