These tests of mother's milk are not useful to determine a confidence interval for the risk of contamination of children. Kjetil explained this very nicely in his answer that we end up with a large confidence interval of [0, 0.5], and AdamO has mentioned in the comments that this actually makes no sense because the binomial distribution is not a good model (below I argue that, while we may not have a clear idea of a better model, we can be reasonably certain that it will make the situation only/likely worse*).
But these tests are not made in order to make some estimate of the probability that mother's milk might be contaminated. Instead, they are used as a search for a Black Swan and to test the theory that babies do not get sick. We have already good reasons to believe that mother's milk is not contaminated, or at least not such that it makes babies get COVID-19 (at least not with heavy symptoms). This test is a trial for that theory. If for some weird reason mother's milk would be heavily contaminated but babies still do not get sick, then this test would have shown that.
It depends on the probability model for the outcome. From a biology perspective, I don't think it makes sense to assume a binomial model.
Which models are alternatives?
It's difficult to pinpoint a particular model, but at least some effects warn us that the binomial model may be inaccurate
- Should we consider the six tests independent? Possibly they have some correlation ,e.g. some testing-problem that simultaneously occurred in all samples. (RNA is unstable, maybe the virus makes it to the baby but not to the lab)
- Is infectivity of breast milk considered a constant? Potentially the breast milk will be more or less infectious depending on the difference in the time that the mother got infected and the time that the breast milk was sampled.
So while there might not be a particular alternative model, we should treat the binomial model with a lot of caution. It is a bit of an ideal postitive situation. Also, the probability that the milk contains the virus is not equal to the probabilitiy that kids get sick from it. If the breast milk of infected mothers only contains virus, say, 10% of the time for each mother, then sampling milk from those mother's may easily give 6 negaŧive results but still, nearly 100% of the babies may get sick
A better measure that we do not need to worry about babies (and breast milk) would be more direct measurements of the outcome (do babies/kids get sick?). Sample a set of babies that have pneumonia for the presence of nCoV-19 virus (or, if the sampling/testing is too expensive or otherwise not possible, then look at the number of cases and see whether it increase like the cases in the elderly population).
In that way, you can see whether the risk of getting pneumonia/SARS increases for babies. What I understand from the stories (but I have not seen raw numbers and I am not sure whether babies are actually actively tested) is that babies do not get the virus.
So, since babies do not get COVID-19 (at least not many), we do not need to worry about nCoV-19 virus in mother's milk (unless we give mother's milk also to the elderly population).
Keep on breastfeeding (as also noted by the RCPCH: the benefits of breastfeeding outweigh the risks) If babies can get sick from nCoV-19 then breastmilk might actually be the thing that protects them and get's them better. We know that breastfeeding is good. But regarding a risk of COVID-19 for babies**... there isn't any data that shows that babies are under considerable risk due to the spread of this new virus. Data, data, data, use real data and not a panic interpretation of the data. Let's not kill the cats to control the plague.
(*)One loophole in my reasoning, that the situation will only be worse when we use alternative models, would be some particular Bayesian models. Say that, based on prior knowledge, the probability that mother's milk is considered either heavily contaminated with probability of contamination p larger than 0.9, or very little contaminated with probability of contamination p smaller than 0.1. Then a posterior distribution for $p$ would lean heavily towards $p<0.1$. So the evaluation of this test with 6 moms may depend a lot on prior knowledge.
The computed posterior is demonstrated below. On the left when we would have a continuous prior for p<0.1 and p>0.9 ,in which case the posterior scales like $p^7$. On the right when we would have a beta distribution as the prior, in which case the posterior is also a beta distribution (and simple to compute by just adding the observations to the parameters of the prior distribution). In both cases a prior distribution that places the probability more to the edges rather than the center (either the contamination probability of breats milk is high or it is low, will make that the result of the 6 cases is stronger than we would imagine from using the confidence interval that has no idea of these prior distribution.
(**) Besides a risk for babies getting sick we actually would also need to consider the risk that babies may not get sick but still pass on the disease. However since babies have not much close contact with many other people this may not be expected to be a main/important route of transmission. And the advise given by the RCPCH (to have other people feed expressed breastmilk to the babies/kids) may actually increase the risk of spread, and get those others sick.