Slow internet speed

The pictures you have provided have confirmed what I said.  ReadyDrop is using a majority of your Surface's avalialble connection speed, not your internet speed.  This is an indication that you have a poor (or poorly configured) wireless connection to your router or just a router incapable of today's normal speeds.

If you don't have a router (or wifi access point) capable of wireless-AC access, it looks like you need one.  Your Surface does have that capability.  If you do have an AC router, then your Surface is not properly negotiating an AC connection to it.  You are getting no better than wireless-G (54Mbps, circa 1997).  Note that obstructions, distance, and interference can be causes for poor connection speed,even if it is wireless-AC, but I doubt that's it based on your posts.

If your router is provided by your internet provider, that can be the cause.  They often only worry about matching the wifi (and ethernet) speed with the internet speed, not caring that your intranet (in your own home, like to the ReadyNAS) speeds will be adversely affected.  Or, they charge more to give you more, and you aren't buying.

Also, if you are using WEP authentication, it will throttle your speed to wireless-B/G.

Note that once you get this aspect fixed, a slow Ethernet connection to the NAS can also slow down the file transfer, but should not adversely affect your ability to do other things on the internet concurrently with your Surface.