It was so ready I thought I must have done it wrong, but I did get a confirmation email saying they got my request. Then I got some documents in the (snail)mail from the new supplier. After applying I still had two months with ES as my supplier still, then on my third bill it was the new one

The funniest part is that the bill (still from Eversource as they're the distributor still) shows me how much I saved that period by not using ES as supplier lol.