router is slow to connect but downloads quickly

You seem to have a mobile UMTS/3G-to-Wifi AP here.
If this is true you might be a victim of the bufferbloat phenomena. There's also an interesting tech talk from Google on this topic.

Search the AP's settings for something related to the above problem.

This relates to response speed as opposed to download speed. Every time your router makes a request, it has to wait for an answer (thereby response) and this usually takes some time on a mobile modem. The request are small, so it is not the size of the request that's taking time, it's waiting for the response. A normal internet site requires multiple requests before it is fully loaded.

However, when you are requesting a download, you only request the download to start. So after it has started, your modem doesn't have to send more requests on behalf of the download, which means that there are no interruptions. That makes your download reach the full speed that's available.