This answered the title, not the OPs actual question. Look to the edit at the bottom for the direct answer to OPs question.
Actually, it could, just indirectly.
Technically the answer is no, because the wireless devices you have (let's assume wireless 802.11N) are all communicating at that protocols fastest speed already. Which can be anywhere between 50-300Mbps.
Now, if your ISP was previously at 10Mbps and you were trying to download something from the Internet, not from another device connected to your LAN, it will be stuck downloading at 10Mbps even though your wireless is capable of higher speeds.
Since you upgraded your ISP speed, let's say to 50 Mbps, you are now able to download that much faster, over the wireless too.
But keep in mind, your wireless speed isn't technically getting faster, just your ISP speed. But because your speed is determined by the slowest point in the network, (in this case your ISP speed) then anything you do over your wireless network accessing the Internet will be faster.
This is only true of course if your ISP is the slowest link in the network. If your devices are performing at their spec protocol speed and that speed is less then your ISP speed, these rules do not apply.
EDIT: after reviewing the question in its entirety, the answer is no. Your iPhones speed is now limited to that speed not because your ISP speed but your wireless setup that you have. Adding a faster, better router might help, testing on a non iPhone 6 might also show different results. It's possible that if you tested this on a full fleged laptop that you might see sub 500mbps speeds. At this point, your ISP speed is absolutely not a factor in getting your wireless devices to talk faster to the router.