First off, you should not test the local internet speed with speedtest.net or similar websites. This just tests the connection you have to your provider in most cases.
Use something like iperf3 instead between two local devices, like your phone and the access point. Check if there is any such tool on your access point and get a coresponding client on your phone.
The wifi speed in general first depends on the generation used. Check which version is the highest commonly supported one between your phone and the access point.
The big second factor is signal strength. How close are you to the access point and what obstructions are in the way?