How to calculate conditional maximum likelihood estimator I have a function , for instance
$$
\lambda(S, B)=\left(\frac{n-(S+B)}{\sigma_{\text {stat }}}\right)^{2}+\left(\frac{B^{\text {obs }}-B}{\sigma_{\text {syst }}}\right)^{2}
$$
maximum likelihood estimator $$ \hat{S}=n-\theta^{\text {obs }} ,~~~\hat{B}=B^{\text {obs }} $$ can be probably taken as $$ \partial\lambda/\partial S= 0, ~~~ \partial\lambda/\partial B= 0  $$
Then, how can I get conditional maximum likelihood estimator $$ \hat{\hat{B}}(S) $$

Just treat $S$ as a known constant and maximise with respect to $B$.  So you have:
$$\begin{align}
\frac{\partial \lambda}{\partial B}(S,B) 
&= -2 \Bigg[ \frac{n-S-B}{\sigma_\text{stat}} + \frac{B^\text{obs}-B}{\sigma_\text{syst}} \Bigg], \\[12pt]
\frac{\partial^2 \lambda}{\partial B^2}(S,B) 
&= 2 \Bigg[ \frac{1}{\sigma_\text{stat}} + \frac{1}{\sigma_\text{syst}} \Bigg] > 0. \\[12pt]
\end{align}$$
This shows that $\lambda$ is strictly convex in $B$, so its maximum occurs at the unique critical point that solves:
$$\frac{n-S-\hat{B}(S)}{\sigma_\text{stat}} + \frac{B^\text{obs}-\hat{B}(S)}{\sigma_\text{syst}} = 0,$$
which gives the conditional MLE:
$$\hat{B}(S) = \frac{(n-S) \sigma_\text{syst} + B^\text{obs} \sigma_\text{stat}}{\sigma_\text{syst} - \sigma_\text{stat}}.$$
(Your value $\theta^\text{obs}$ does not appear in the original objective function and is not defined, so it is not clear where this came from in your question.)
