I am in the process of making this cake now for my son's birthday party this evening. I made the custard and had the same exact thought you did - this is sure runny! I beat it a little longer than 2 minutes, thinking - like you, that it might thicken more. I noticed it had a huge amount of froth on the top. It also appeared to be more like a charlotte rousse than a custard, which is usually think by the time you finish mixing it.
I was very tired last night after I finished making fondant and was on the fence about making the custard last night or this morning. I looked online, via a search, and found nothing about whether the custard needed to be at room temperature, cooled, etc. so decided to wait and make it this morning. NOW I found a different printing of his recipe which says the custard has to cool in the fridge for 6 hours!!
Our party is at 5:30, but we have to leave our house by 4:30. I don't think there's any way I have time to let the custard sit for 6 hours, bake the cakes, cool them, assemble, crumb, frost and finish with fondant!! I'm panicking because I really don't want to make the cake without the custard but don't have the 6 hours to wait.
I'm hoping that it will thicken enough to use in less time.
Anyway - just wanted to let you know I had the same experience/thoughts regarding the thickness of the custard, so it's not just you.