FYI I did this today at the Thai Embassy in London it only took 15 minutes. I called in advance and no appointment was needed. It's done in the basement part of the building.
The only document you need to take is your own passport and a stamped addressed envelope for return of the consent letters (You get one per child)
Then you fill in the consent form which you showed (although they use a hybrid one with English too) and a payment form. Basically just add your name address and names of the children.
Then they photocopy your passport get you to sign it and pay their fee which is £10 (that was the total for 2 children)
So in a few days I'll get the verified consent letters posted to me.