If you have two router devices that all your network has to traverse through, then yes you have a double NAT which isn't always a bad thing (I've had a double NAT at a few offices without problem going through a modem/gateway and then through the wireless router) but it's not as efficient.
One of the key things to look at, if your computers are connecting together within your LAN for sharing files or accessing a server/share, is how many device hops and bandwidth does each computer have to get to the shared device. For instance, if you have a file server connected to a gigabit switch, which also has several workstations connected to the switch, then those directly connected workstations can have up to a gigabit of throughput. However, if you have some wireless workstations, they're only going to get the maximum throughput of the wireless router, such as 54 Mb/s split amongst all the currently connected devices (and that's in an ideal situation, again.)
Usually, when people are thinking of slow internet issues, really it's just the number of devices that they have on their internal network concurrently accessing a limited bandwidth coming from your internet provider. That's always the largest bottleneck in any internet-connected network. If you have sixteen devices accessing to the internet at the same time, you have to take your ISP's rated speed and subtract between 10% and 20% of that for variation in load and quality, then divide that result by the number of computers.
So, at our sign office we only have a 3 Mb/s download speed DSL line, which in reality gets about 2.8 Mb/s throughput. We have 5 users accessing the internet regularly, so each user is getting about 0.56 Mb/s of throughput.
This is the theory anyways. Of course, loads will vary as will when people are accessing information online, but there's generally a lot of transfers happening even if the user isn't aware they are sending or receiving information online. Generally, when I have customers that are having problems with slow internet like this, the results of what they're looking for is just to have a faster internet connection from their ISP, nothing in their internal network will facilitate that faster internet speed except their internet connection.
Now, of course there could always be other explanations. The first is malware, causing a lot of network traffic from certain computers. I've also seen this happen, where a virus was running on a couple computers in the network and sending and receiving massive amounts of traffic which slowed down the whole network. The other possibility could be a lot of other internal network commotion, such as broadcast storms. This is usually caused when you get multiple switches connected together or connect two switches with more than one ethernet cable.
It's not always the best solution, but for a simple network I'd recommend getting a wireless router that has at least four gigabit LAN ports. Connect one gigabit capable switch to one of the LAN ports on the wireless router, and never connect or daisy chain additional switches off of the switch itself, but instead connect additional switches directly to the back of the wireless router. This helps to eliminate broadcast storms or possible network loops because you're using a router to connect and communicate between the switches. But again be careful, if you use just any regular wireless router, its most likely not going to have a built-in gigabit switch, just a 10/100 switch which can cause a massive bottleneck if you have multiple computers trying to pass data back and forth between two switches connected to the router.
