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"Individuals with high LDL cholesterol levels increase their risk of developing coronary heart disease." Would it be "their risks"? Does each individual have a separate risk...therefore "risks"? Or do they collectively have a risk?

Similarly, "Most Americans eat more than enough protein in their diets." Would it be diet? Do Americans have separate diets or a collective diet?

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I would go with the singular for both, I think: "their risk" and "their diet". Although we are talking about a group of people, we are specifically talking about a hypothetical singular example. For instance, we aren't comparing several peoples' diets, we are referring to the singular diet of the average individual.

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I think this is a rather interesting question. I answered a similar question a while ago, about children able to move on their backs.

NGram showed about 10% of references to on their back[s] used the singular, rising to 25% with on their chest[s]. I don't strongly favour either form, but I felt that if up to 25% used the singular that was enough to say it was at least "acceptable" usage. No-one upvoted my answer though.

The arguments here are exactly the same as I advanced before - briefly, each individual has a single risk; it's largely irrelevant that there may be many individuals.

The usage figures are considerably different, probably because people are quite well aware that it's actually the same risk applying to multiple individuals. Well over 90% of written usages over the last 50 years are for the singular. It's perverse in the extreme to argue against any usage so dominant. By my lights, "correct" usage is whatever the vast majority actually say. Semantics, grammar, and logic are only relevant when considering whether a minority usage should be labelled "acceptable", "non-standard", "incorrect", etc. Anyway, here's the picture...

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I will admit the preference for the singular falls to "only" about 65% when it comes to in their diet, but that still looks like a majority to me.

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I think you've already nailed the point. Each individual has a separate risk, so the original example would be "Individuals with high LDL cholesterol levels increase their risks of developing coronary heart disease."

I think we feel that "they"/"their" agrees with "risk" because of the abundance of times that people incorrectly use forms of "they" as a singular pronoun.

If you really want to use the singular form of "risk", then I think the only safe route is to actually make everything singular: "An individual with high LDL cholesterol levels increases his/her risk of developing coronary heart disease."

Also, note that this example would also make sense with the collective sense of "risk", as long as that really is the intended meaning.

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