Your first post had me confused. The title says you have button quail and the first post says they are coturnix quail - an entirely different species. In the case of coturnix quail, the male should indeed not be left with mother and chicks. With buttons, you need to watch him as some males lack the instinct of how to deal with chicks, but from my experience most males will actually actively help raising the chicks and can easily be left with them.
I'll assume your quail are indeed buttons.
The one in the last picture really doesn't look like it's going to make it, but it baffles me - I've never had a button older than a week die on me (not counting the adult that was eaten by a cat), from probably around 20-30 natural hatches.
He looks like he is really sick - he doesn't look cold, and if he couldn't eat he wouldn't have survived this long.
I've never had a heat lamp on any of my natural hatches, the mothers (and in most cases, the fathers as well) have been perfectly able to keep the chicks warm with the surrounding temp being probably between 60 F (maybe as little as 50 F) and 72 F.
The chicks that have died on me, have done so because they didn't get to mom or dad before they got too cold. Since realizing this, obstacles in the cage have been banned from the time the chicks hatch till they are a couple of weeks old. Obstacles being things like straw - I might be too picky here, but they really are tiny when they hatch; I think straw is too rough as bedding - dust baths, hutches, food and water dishes with high edges, tunnels and so on. Since banning obstacles I've lost very few chicks.
I don't think I can help you save the chick that's sick now, but for the future I'd use a different bedding (fine wood chips, shavings, sand (not recommended for brooders the first few days as some chicks tend to think it's food, but it should be fine when mom is there to show them what to eat - I've used it with success), hemp bedding and so on), and as soon as the chicks hatch I'd remove all other 'obstacles' in the cage.
I'd also give the cage a good clean once the current chicks are out of there (they shouldn't be left with the parents for more than 6 weeks or so - if there are no roos among them you will probably be able to get away with it for several months, but I'd still sell/give away the chicks to prevent inbreeding). If whatever causes them to be sick is contagious, you don't want the next clutch to catch it as well.
I also wouldn't use a heat lamp next time, assuming mom is doing her job - I hadn't though of the heat creating drafts, but I guess it's plausible.
Good luck - don't give up!