This looks less like a simple reinstall miss and more like a client sync or cache problem. Microsoft’s current guidance for stale or missing chat updates is to restart Teams and, if that only helps briefly, clear Teams cache. That fits your symptom where mobile updates first and desktop catches up hours later.
The two highest value steps left are to force an update and restart Teams on desktop, then do one clean comparison in Teams on the web. If web chat is instant while the desktop app still lags, that strongly points to local desktop state. If desktop, mobile, and web all lag for both of you, the docs do not show a reliable end user fix beyond cache reset and updating, so it’s reasonable to treat that as a Microsoft-side issue or account-side problem rather than something you can fully repair yourself.