Technically you got to it in time and sucked up all the 'grey' water, you are correct uplift the damaged area of carpet and remove the underlay, you should try and at least rinse the carpet so all of it is wet so you dont get a tide mark.
You should put a good sized air mover or fan on it to speed up the drying and spray the backing with an anti microbial so it doesnt smell.
Unless the carpet was absolutley sodden it should be ok, look out for delamination, if it has secondary backing it may well seperate from the face, then it will be knackered.
I did hundreds when I was with SM, as above it should be ok. Just replace the damaged underlay area (its not supposed to get wet) and refit, then give it a good clean to balance it all.
Tell the builders what it will cost roughly and agree works. DONT DO IT UNLESS THEY AGREE. Give the Invoice to the in laws to pass on. Then cross your fingers they will pay up! If that doesnt work claim on inlaws insurance, the Ins Co will pay you, and they will chase builders for the money. The builders excess will be £500 or so, so its cheaper for them to pay you and not claim.
Good luck.
Murky