In order to use Whatsapp Web you need to be on the same wifi network as your phone.
Your phone has the data on it. Intranet speeds on most wifi networks these days is faster than 50Mbps. That's more than fast enough to transfer all your messages and photos with little to no lag.
It doesn't matter how fast your connection to the WWW is because you're not using the WWW to transfer the data, you're using the wifi intranet.
Yes, that's why I think so. They easily loads all the messages and images in just few seconds even when my phone's internet is too slow.
They won't let you use the web app when you're phone is disconnected, so there's no way to confirm it.