How quickly can a cat regain lost weight?

My friend and I rescued a cat who had been abandoned in an apartment when the tenants left. He was underweight, not terribly so as he still had some fat between his rear legs on his belly and was 9 lbs. However you can feel his spine when you pet him and he feels thin, you run your hand down his body and he is super narrow and his sides suck in drastically after his ribs. The vet gave him a clean bill of health, no concerns in her mind as to his underweight status being due to health issues. Most likely just due to his history (the previous owners did leave a bag of food out for him to eat but I don't know if it was a good food). 

We have had him since Oct 7th and I am not seeing much weight gain, he eats like a monster and loves his wet food (Natural Balance cans 2x a day) and does eat his dry food (grain free Earthborn Holistic). Is a month too soon to be seeing an improvement? His coat also seems to be greasy still, softer than when we got him but still not in great shape. What is a normal time line to see improvement before we bring him back to the vet to make sure he doesn't have other issues? 

He also is very active, vet said he was 3-4 years old and he is very playful and seems happy. 

Since he is eating well, been vet checked and is active I wouldn't be concerned about it yet.  It may just take some time.  He also may be a kitty that naturally has a long thin build, just as some people have a very slight build so that they always look very thin.

Cats really need their protein.  You could add the occasional cooked egg white and raw yolk, both for protein and good fats. 

Watch his elimination habits as well.  A lot can be told from the litter box.

Other than that, give it a little more time...  I'd say 'bout a month.

Sounds odd to me.   If he is eating like a monster then I would expect the protruding spine to disappear in a week at most.   To tell you the truth from your description this cat sounds hyperthyroid to me.   Did the vet test for that?  Probably not.  

That sagging under his tummy isn't fat, some cats have that saggy, swinging skin but not on their bodies.

Your description makes him sound very thin.

Baby, a gravely sick cat I rescued was a large framed cat, felv positive, and I rehabbed her in 4 months, from 4 lbs to nearly 10. Her fur was renewed and crazy soft and healthy then too.

Nutrition, high calorie foods, and plenty of love and attention should do the job

FYI I fed Baby Babycat by Royal Canin dry food and canned daily.