Each child can be modeled as a Bernoulli r.v. $X_i$ with probability of having the disease equal to $p_i$, $X_i \sim B(p_i)$, $i=1,\dots ,n$. If you assume that a) $p_1 =p_2=\dots=p_n=p$ and b) that these are independent rv's then their joint density is
$$f(X_1,\dots,X_n) = \prod_{i=1}^{n}p^{x_i}(1-p)^{1-x_i}$$
and their log-likelihood function, viewed as a function of $p$ is
$$\ln L =\sum_{i=1}^{n}\left\{x_i\ln p+(1-x_i)\ln (1-p)\right\}$$
which leads to the MLE for $p$
$$\hat p =\frac 1n\sum_{i=1}^{n}x_i$$
which is unbiased since $$E\hat p =\frac 1n\sum_{i=1}^{n}Ex_i = \frac 1n np =p$$
Consider now the variable
$$U_i = X_i - E(X_i) = X_i -p \Rightarrow X_i = U_i + p$$
We have
$$EU_i = 0,\qquad Var(u_i) = Var(X_i) = p(1-p) $$
so it is covariance-stationary.
Subsitute for the $x$'s in the estimator
$$\hat p =\frac 1n\sum_{i=1}^{n}(u_i+p) = \frac 1n\sum_{i=1}^{n}u_i +p$$
and consider the quantity
$$\sqrt n (\hat p-p) =\sqrt n\frac 1n\sum_{i=1}^{n}u_i= \frac {1}{\sqrt n}\sum_{i=1}^{n}u_i$$
Since the $U$'s are covariance stationary, (and evidently i..i.d) then the CLT certainly applies and so
$$\sqrt n (\hat p-p) \rightarrow_d N\left (0, p(1-p)\right) $$
For approximate statistical inference, we manipulate this expression through
$$ \sqrt n (\hat p-p) = Z \Rightarrow \hat p = \frac {1} {\sqrt n}Z +p$$
and write that, for "large samples"
$$\hat p \sim_{approx} N\left (p, \frac {p(1-p)}{n}\right)$$
(but not when $n$ truly goes to infinity, since then $\hat p$ does not have a distribution, but collapses to a constant, the true value $p$ since $\hat p$ is a consistent estimator).
