This is the worst thing about mu/mu4e. You need to do the indexing with a timer in Emacs, and because Emacs is single-threaded this blocks things / slows down the UI. After trying to get it to work in some automated way I gave up. I manually issue a key-combo to re-index and then read my E-Mail.
Another comment: I recommend not using Emacs to send E-Mail, but to outsource this to a local smtpd such as Exim (which I use with mu4e on Debian). Then if you don't have WiFi or whatever the smtpd just queues those up for later sending, and has queue inspection tools etc. All of the solutions for doing this in Emacs itself are dirty hacks full of edge cases and bugs smtpds have solved decades ago.
Despite some rough edges, mu+mu4 (plus Exim, or similar). is an amazing tool, and the most useful E-Mail client I've ever used.