To answer my own question, I'll try a Fermi estimate with what I know so far. These numbers are based on what I remember hearing and may be very inaccurate. It's better to react too early than too late, so I'll be using a more pessimistic model.
If anyone has better numbers, or spots an error or has a good reason to change the formula, please chime in with a comment, and I'll consider edits.
There are about 3 hospital beds per 1000 people in the United States. Normally, about 2 of those are occupied, leaving 1 free per 1000 people to deal with the virus. Therefore, when the number of severe cases exceeds a "critical mass" of 0.1% of the population in your area, the medical system becomes overwhelmed, and cases become more deadly.
I think about 80% of cases are supposed to be mild, so 20% may require a hospital bed. Therefore, when the total number of cases in your area hits 0.5% of the population, we hit the critical 0.1% mark of severe cases.
Lukas_Gloor estimates that at best 5% of cases are diagnosed in the US. Therefore, we hit critical mass when the number of reported cases is projected to hit 0.025%. That's about 250 confirmed cases per million people in your metro area.
But by that point it's too late. You do not want to have a case of pneumonia when we hit critical mass. But it takes time for a case to progress that far. Lukas_Gloor estimates maybe 20 days for symptoms to progress to pneumonia, plus 7 days incubation, so you need to isolate 27 days before critical mass.
Lukas_Gloor estimates a doubling time of 2.5 days. That means about 11 doubling periods to add up to the 27 days. 11 doublings is 2^11 = 2048. Projecting 11 doubling periods backwards from 250/M, it should be about 27 days before critical mass, when there are about 250/2048 confirmed cases per million or about one confirmed case per ten million.
Conclusion: if your US metro area has less than ten million people, then isolate on the first confirmed case in your area. Otherwise, isolate when there is at least one confirmed case per ten million.
Am I being too pessimistic? This does seem too early, but so goes exponential growth. Catching the virus after this point would be very bad, but there would only be about 20 people per ten million who could actually give it to you that soon. What are the chances of catching it? Maybe they're too small compared to risks you take every day (like driving a car to work) to warrant isolation that early.
When we hit the projected critical mass of confirmed cases, how many of them will have progressed to pneumonia? Maybe there's a bit more time before the system is overwhelmed. On the other hand, at what stage of the disease do people get diagnosed? If they're already severe by then, this might be moot.
I was surprised by recent accelerations in numbers and also by the estimate on how large the difference between true cases and reported cases is in certain places. I think there are many locations in Europe and the US where the best time to go into strong isolation would have been 7 days ago. I don't think there are many large cities in Europe where going into isolation now is too early.
I think 20 days before peak hospital crowding has a good shot at being safe, but it's not a guarantee because incubation may take 7 days (in rare cases even longer?) and as you say, developing severe pneumonia may also take quite a while (especially in healthy young people?).
I think there's a weird effect where by the time when you want to go into isolation based on timeline-of-the-illness considerations, so few people in your community will be infected that it seems too early. But by the time your chances of becoming infected go from negligible to not-entirely-negligible, it's already straining the timetable in the instances where you get unlucky.
How few? Ten confirmed cases? A hundred? Does it depend on the size or density of the community?
The way I thought about it, I started to take risks somewhat seriously by the time 1 in 1000 in my neighborhood might be infected (and then realized that this was already pushing it with the timing-of-illness considerations!). If you're in a country where testing is very diligent, then diagnosed cases might be 12% of actual cases. If you're in the US, it's 5% at best. It's also important to know when countries switch from a policy of testing everyone who might plausibly have the disease (if they ever had such a policy) to a policy of only testing people with pre-existing health conditions or severe symptoms. In the latter case, estimating the progression of the outbreak from past doubling times might be better than just continuing to follow the numbers. (Although, scarily enough, the numbers might continue to grow similarly fast even after this switch of policy.) I used 3.5 days as the doubling time initially, but I think that's what led to me being a bit late despite the intention of wanting to be comfortably early. Playing it safe, I think 2.5 days might be better. Of course, things might change especially once governments start shutting down a lot of things.
A friend also gave me good advice about not going to the grocery store I usually go to, because she realized that it was right next door of one of London's larger hospitals. That was a useful protection measure 20 days ago already.
"It's hard to time your reaction to an exponential process, so the only two choices are to act too early or act too late."