Anecdotal evidence suggests that it is fairly common: https://www.reddit.com/r/COVID19positive/ -- 2 of the 5 top posts from today are from people complaining about experiencing this, and are both full of comments personally relating to it. There is obviously going to be a selection bias here, but it seems like a good starting point for estimating a lower bound if you can't find enough good studies.
This study finds about a third of patients self-report fatigue at 2-3 weeks after a positive coronavirus test. That might be a reasonable generous upper bound for the likelihood of chronic fatigue, though I wouldn't trust telephone studies of self-reported symptoms all that much.
Fatigue that lasts 2-3 weeks after the worst symptoms are over is common with essentially all bad viral infections - post-flu fatigue is common for example (can't find any good statistics on how common). So, I don't know if 1/3 reporting fatigue 2 to 3 weeks after tells us anything useful about how common post-covid fatigue lasting months afterwards is
These are patients who had a positive test in April. Most infected people without symptoms or with mild symptoms did get tested in April in the US. We know about 20-40% are asymptomatic, with higher % among younger people. So actual rate based on this study would be upper bounded by 1/4 (not 1/3) and point estimate closer to 1/5. (I also agree with SDM).
From a Nature news article last week:
One study7 of 143 people with COVID-19 discharged from a hospital in Rome found that 53% had reported fatigue and 43% had shortness of breath an average of 2 months after their symptoms started. A study of patients in China showed that 25% had abnormal lung function after 3 months, and that 16% were still fatigued8.
https://institute.global/policy/long-covid-reviewing-science-and-assessing-risk
From the COVID Symptom Study in the UK (app based questionaire), "10 per cent of those taking part in the survey had symptoms of long Covid for a month, with between 1.5 and 2 per cent still experiencing them after three months", and they claim "long Covid is likely a bigger issue than excess deaths as a result of Covid, which are between 0.5 per cent and 1 per cent".
App-based survey, so not necessarily representative of population. Not clear how severe the 3 month cases are, though they state "The most common reported symptom has been described by doctors as “profound fatigue”". Article also summarizes other related studies.
Conclusions and Relevance In this study of a cohort of German patients recently recovered from COVID-19 infection, CMR revealed cardiac involvement in 78 patients (78%) and ongoing myocardial inflammation in 60 patients (60%), independent of preexisting conditions, severity and overall course of the acute illness, and time from the original diagnosis. These findings indicate the need for ongoing investigation of the long-term cardiovascular consequences of COVID-19.
While this measurement isn't CFS directly the heart damage might be the causal mechanism for it and is thus worrisome news.
Found some evidence about fatigue in the linked paper
Compared with pre–COVID-19 status, 36 patients (36%) reported ongoing shortness of breath and general exhaustion, of whom 25 noted symptoms during less-than-ordinary daily activities, such as a household chore.
That said, recently recovered patients are not a particularly helpful group to look at (the average period since diagnosis was ~70 days, which might mean they'd generally been recovered for less than a month)
Coronavirus: How long does it take to recover?
The fever should settle in less than a week, although the cough may linger. A World Health Organization (WHO) analysis of Chinese data says it takes two weeks on average to recover.
Given that of those 25 only 6 got hospitalized (and thus might have needed more recovery time) I think the recovery time is likely higher then a month and nearer to two months.
The initial paper also has some plots that show measurements of myocardial inflammation against the time since infection and those plots don't suggest that the time has a huge influence. The native T1 plot even looks like the amount of myocardial inflammation rises as time passes.
https://www.microcovid.org/paper/2-riskiness#fn6 discusses https://covid.joinzoe.com/post/covid-long-term which has an app-based survey claiming 1 in 10 people still have symptoms after 3 weeks. (but since people can just sign up for the app I'd guess this is harder to know how to interpret than the telephone survey). Microcovid.org uses this 1 in 10 figure as the estimate for chance of some ongoing health consequence, and claims the risk of ongoing health problems from a 1% chance of COVID is equivalent to the risk from 1 year of driving (but this comparison involves even more assumptions).
Not addressing fatigue, and just a study in progress, but this study is looking for long term neurological problems, might another weak bit of evidence when it releases results https://www.cambridgebrainsciences.com/studies/covid-brain-study
Months and months ago, when COVID-19 first broke, one of the most concerning aspects of the disease was the possibility that it might produce long-term chronic fatigue, based on a comparison with SARS (apparently a large percentage of the people who had SARS in 2003, had chronic fatigue symptoms years later, though I can't find the paper right now), plus some other evidence.
At the time, we didn't have much data, but now we're a few months into the pandemic. Obviously, we won't know how long lasting it is, but what are the updated risk estimates of chronic fatigue from COVID?