We are using mainly Android devices to access Nextcloud. However, I have also an iPhone 7 and tested out of curiosity the iOS client. I found the iOS experience by far better than on Android. On my Pixel, it takes a few seconds to display search results once I pressed enter. On the iPhone results come nearly instantly while typing. To my understanding the search takes place on my server and only the results are pushed to the clients.
Is there something I can check or tweak to speed up operation on the Android device?
There is nothing you can do at the moment 
Android is triggering server side searches and displays the results while the actual Nc13 API doesn’t support limited search queries so we get all hits at once. The iOS app isn’t triggering server side searches but massively caches the search results in the background and searches its local image index. Thus the iOS client seems faster but works completely different. With Nc14 (or later, not sure) the search API of the server will allow us to receive smaller result block and thus will speed up the search on the Android client.