I want to know, if I purchase ACF PRO (one site) and use it on my WordPress site hosted on WP Engine, which has three environments: development, staging, and production, will this one license work for all environments or only for one?
You should have no issue with this, .local is recognized as a dev environment and my staging environments are all recognized as staging in my ACF panel. Then my production sites are properly recognized as well.
This is from the FAQs on the pricing page.
https://www.advancedcustomfields.com/pro/#pricing-table
Development and staging sites are ignored and do not count towards site activation limit. For the full list of rules we apply when detecting a development or staging site, please see our License Activations guide.