I'm a YouTuber, I've licensed dozens of my videos, to dozens of different companies. This guy has no idea what he is talking about.
Firstly, at 1:10 he says, "They came back with an offer to pay me through paypal" and then the YouTuber says he said "okay". You now have an implied agreement and informal written contract. The Weather Channel has informally been granted the right to use his material, not all agreements need to be formal. Also, since The Weather Channel went ahead and used it, they now legally are required to pay him whatever fee they agreed upon.
Companies usually pay on net 30 (30 days) from either: executed license agreement or air date. Some companies will pay net 60. Yes, so let's say America's Funniest Home Videos contacts you about licensing that cute dog video you uploaded that went viral, you might not get paid until a month after it airs, which could be 3 months or longer from the date they contact you. Honestly, the guy just seems like he doesn't know how licensing works (which is surprising, because given his video history I'd expect he has had licensing inquiries in the past). These are usually big companies, and believe it or not, they have a way and set guidelines and procedure for how they pay people - and this almost never includes paypal payments. They also need tax information and an actual material release.
Also, unless it specifies in a contract or in their informal agreement that he would be paid right away, there is no obligation by The Weather Channel to do so. The Weather Channel must make payment in a "reasonable amount of time" - which is probably in line with their standard payment procedure (which I presume is net 30 or on their next payment cycle).