I went through menopause almost 6 years ago; right about the time I was going from a casual cyclist to a more intense one. I never noticed any effects on my training. I had hot flashes, but never while riding. They mostly were at work, when I was in a hot environment, or at night, when sleeping.
I just took a big drink of cool water to stop them.
The one thing it did compromise was my comfort on the saddle. After using the same one for a couple of years, the vaginal dryness started bothering me. I got a new saddle and went to my doctor who prescribed an E-string, which emits a miniscule dose of estrogen locally, only. You wear it for 3 months and then get a new one. It takes a few months for it to work, but it really works!
You might feel more tired and need a longer recovery time than previously, but that is more a function of aging than menopause.