Let $t=T_F$. Keeping in mind that $t_1,t_2,\dots,t_N$ are unif$(0,t)$ given the number of occurrances $N=n$, the likelihood becomes \begin{align} L(\lambda,t) &= P(N=n) f(t_1,t_2,\dots,t_N|N=n) \\ &= \frac{e^{-\lambda t}(\lambda t)^n}{n!}(\frac1t)^n \\ &\propto e^{-\lambda t}\lambda^n. \end{align} for $t\ge t_\text{max}$ and zero elsewhere. This is maximised for $\hat t=t_\text{max}=\operatorname{max}(t_1,t_2,\dots,t_n)$ and $\hat\lambda=n/t_\text{max}$. These MLEs don't exist if there are no occurrences $N=0$, however. Conditional on $N=n$, using the fact that the $t_i$'s are uniform, $E(t_\text{max}|N=n)=\frac n{n+1} t$. Hence, the estimator $t^*=\frac {n+1}n t_\text{max}$ is unbiased conditional on $N=n$ and hence also conditional on $N\ge 1$. A reasonable frequentist estimator of $\hat\lambda$ might be $\lambda^* = n/t^* = \frac{n^2}{(n+1)t_\text{max}}$ but this does not have finite expectation when $N=1$ so assessing its bias is even more troublesome.
Bayesian inference using independent, non-informative scale priors on $\lambda$ and $t$ on the other hand leads to a posterior $$ f(\lambda,t) \propto e^{-\lambda t}\lambda^{n-1}t^{-1}. $$ for $t>t_{max},\lambda>0$. Integrating out $\lambda$, the marginal distribution of $t$ becomes $$ f(t) = \frac{n t_\text{max}^n}{t^{n+1}}, t>t_\text{max}. $$ with posterior mean $E(t)=\frac n{n-1} t_\text{max}$. A $(1-\alpha)$-credible interval is given by $\left(\frac{t_\text{max}}{(1-\alpha/2)^{1/n}}, \frac{t_\text{max}}{(\alpha/2)^{1/n}}\right)$.
The marginal posterior of $\lambda$, \begin{align} f(\lambda) &\propto \int e^{-\lambda t}\lambda^{n-1}t^{-1} dt \\ &= \lambda^{n-1}\Gamma(0,\lambda t_\text{max}) \end{align} where $\Gamma$ is the incomplete gamma function.