I'm not sure if this helps, but I know that Hugh sang it on his tv show "Jeeves and Wooster". A link is here:
link

Given this, I would say it's very likely that he's the one singing, but I'm not really sure.

thanks

I know I'm, like, 3 months behind in replying to this forum, but I think it is Hugh Laurie singing, but it kinda sounds like he is singing in his American accent...

I know what you mean. I think it is Hugh Laurie but I don't know if I think that because I want it to be him or if it actually sounds like him. Plus I am sure there are some parts that are him but I can't tell if he is singing the whole thing or not...

I'm 99.9% sure its Hugh. At first, I was a bit skeptical, but I watched all the "behind the scenes" footage on youtube from making the two studio songs, and it was either Hugh or Bob Guiniey. They showed them each singing the song, and I can tell you one thing's for certain, its not Bob, lol!

Here's the link, love watching this:
link

Hugh sings around 2:27, you can hear in his voice when he sings the "Red HOOOT hoochie coocher", the "Hot" inflection is the identifier, its Hugh :D

It was Hugh on lead vocal. Primarily his arrangement, too. With Lester Holt doing the newsman voice and playing bass. Bonnie Somerville doing the improvised female vocals.