Asymptotic Standard Error of Estimator that is a Function of Two Samples 
Suppose that $X_1,...,X_m$ and $Y_1,...,Y_n$ are independent exponential random variables with $X_i\sim EXP(\lambda)$ and $Y_j\sim EXP(\theta \lambda)$.  This implies that $E(X_i)=1/\lambda$ and $E(Y_j)=1/{\lambda\theta}$.
Let $\hat\theta_{m,n}$ be the maximum likelihood estimator of $\theta$.  Find an expression for the approximate standard error of $\hat\theta_{m,n}$.  Assume that $m$ and $n$ are large and use asymptotic theory.

I end up calculating that the MLE is $\hat\theta_{m,n}=\frac{\bar X}{\bar Y}$, where X and Y bar are just the sample means of the respective distributions.  I'm not one hundred percent confident in this, but it seems right.
So from here, I need to find the variance of this expression; particularly, the limiting variance; but here's the issue; this MLE depends on both $m$ and $n$!  In my experience, I've never had to manipulate an MLE like this; so the typical way to do this (when you have just one sample and one parameter):
$$\sqrt{n}(\hat\theta-\theta)\to^d N(0,I_1^{-1}(\theta))$$
doesn't make sense any more!  Or at least, not to me.  So how can I possibly "find an expression for the approximate standard error of $\hat\theta_{m,n}$?"
 A: Checking the MLE: From your specification of the problem, your log-likelihood function is:
$$\begin{equation} \begin{aligned}
\mathcal{l}_{\boldsymbol{x},\boldsymbol{y}}(\theta, \lambda) 
&= \sum_{i=1}^m \ln p (x_i | \lambda) + \sum_{i=1}^n \ln p (y_i | \theta, \lambda) \\[8pt]
&= \sum_{i=1}^m (\ln \lambda - \lambda x_i) + \sum_{i=1}^n (\ln \theta + \ln \lambda - \theta \lambda y_i) \\[8pt]
&= m ( \ln \lambda - \lambda \bar{x} ) + n ( \ln \theta + \ln \lambda - \theta \lambda \bar{y}).
\end{aligned} \end{equation}$$
This gives the score functions:
$$\begin{equation} \begin{aligned}
\frac{\partial \mathcal{l}_{\boldsymbol{x},\boldsymbol{y}}}{\partial \theta}(\theta, \lambda) 
&= n \Big( \frac{1}{\theta} - \lambda \bar{y} \Big), \\[8pt]
\frac{\partial \mathcal{l}_{\boldsymbol{x},\boldsymbol{y}}}{\partial \lambda}(\theta, \lambda) 
&= m \Big( \frac{1}{\lambda} - \bar{x} \Big) + n \Big( \frac{1}{\lambda} - \theta \bar{y} \Big).
\end{aligned} \end{equation}$$
Setting both partial derivatives to zero and solving the resulting score equations yields the MLEs:
$$\hat{\theta}_{m,n} = \frac{\bar{x}}{\bar{y}} \quad \quad \quad \hat{\lambda}_{m,n} = \frac{1}{\bar{x}}.$$
(Note that in the case where $\bar{y} = 0$ the first of the score equations is strictly positive and so the MLE for $\theta$ does not exist.)  This confirms your calculations of the MLE.

Adjusting the MLE to remove bias: Treating the MLE as a random variable we have:
$$\hat{\theta}_{m,n} = \frac{n}{m} \cdot \frac{\dot{X}}{\dot{Y}},$$
where $\dot{X} \equiv m \bar{X} \sim \text{Gamma} (m, \lambda)$ and $\dot{Y} \equiv n \bar{Y} \sim \text{Gamma} (n, \theta \lambda)$ are independent random variables.  From this equation, the MLE is a scaled beta-prime random variable:
$$\hat{\theta}_{m,n} \sim \theta \cdot \frac{n}{m} \cdot \text{Beta-Prime}(m, n).$$
This estimator has expected value $\mathbb{E} (\hat{\theta}_{m,n}) = \frac{n}{n-1} \cdot \theta$, which means that it has positive bias.  We can correct this bias by using the bias-adjusted MLE:
$$\tilde{\theta}_{m,n} = \frac{n-1}{n} \cdot \frac{\bar{X}}{\bar{Y}} \sim \theta \cdot \frac{n-1}{m} \cdot \text{Beta-Prime}(m, n).$$

Standard-Error of the adjusted MLE: The adjusted MLE is unbiased.  It has variance:
$$\begin{equation} \begin{aligned}
\mathbb{V}(\tilde{\theta}_{m,n}) 
&= \int \limits_0^\infty \Big( \theta \cdot \frac{n-1}{m} \cdot r - \theta \Big)^2 \text{Beta-Prime} ( r | m, n) dr \\[8pt]
&= \theta^2 \cdot \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)} \int \limits_0^\infty \Big( 1 - \frac{n-1}{m} \cdot r \Big)^2 r^{m-1} ( 1 + r )^{-m-n} dr \\[8pt]
&= \theta^2 \cdot \frac{n+m-1}{m(n-2)}.
\end{aligned} \end{equation}$$
The corresponding standard error is:
$$\text{se}(\tilde{\theta}_{m,n}) = \tilde{\theta}_{m,n} \cdot \sqrt{\frac{n+m-1}{m(n-2)}}.$$
Letting $\phi \equiv m/n$ and taking the limit as $n \rightarrow \infty$ we obtain the asymptotic approximation:
$$\text{se}(\tilde{\theta}_{m,n}) \approx \frac{\tilde{\theta}_{m,n}}{\sqrt{n-2}} \cdot \sqrt{\frac{1+\phi}{\phi}}.$$
This gives you both exact and approximate expressions for the standard error.  I hope that is helpful.  (Please make sure to review my algebra to make sure I haven't made a mistake!)
