Suppose we have a pie $Lv=0$ and we approximate it using the scheme $L_k^n u_k^n = 0$
Suppose that this scheme convergence to the solutions so we have that $|| Lv - L_k^n u_k^n || \to 0 $ as $\Delta t, \Delta x \to 0$ according to the definition.
Now obviouslyy if $\Delta x= \frac{1}{N}$ where $N$ is the number of nodes - 1 then the as $N$ gets large one gets better approximation. so we see that the convergence order is different for different $N$.
My question is: How do we calculate the convergence order for different values of $N$?
In general in numerical mathematics, a numerical method is said to have order of convergence $r$ if for some constant $C > 0$ independent of $N$ $$ \|y - y_N\| \leq CN^{-r}, $$ where $y$ is the exact solution to some given problem (in your case, $y = Lv$) and $y_N$ is an approximate solution to the same problem (in your case, $y_N = L^n_ku^n_k$ - notation I don't fully understand by the way, what is $n$ and $k$?), and $N$ indicates some degree of refinement employed for computing the approximate solution.
Hence, it doesn't make much sense for me to talk about different order of convergence for different values of $N$, because actually the order of convergence is fixed independently of $N$.
How does one estimate it numerically? If you have a problem whose solution $y$ is known, that is a rather easy task. You compute the approximate solution $y_{N_0}$ for some $N_0$, double it to $2N_0$, compute $y_{2N_0}$. Then, the number $r$ will be well-approximated by the base-2 logarithm of the ratio of the errors you found. How? Since $C$ is supposed to be independent of $N$, we have $$ \frac{\|y - y_{N_0}\|}{\|y - y_{2N_0}\|} \approx \frac{2^r N_0^r}{N_0^r} = 2^r \implies r =\log_2 \left(\frac{\|y - y_{N_0}\|}{\|y - y_{2N_0}\|}\right). $$
If you don't have the exact solution $y$, the standard practice is to compute a reference solution with a $N$ much larger than $N_0$, and employ this as a surrogate of the exact solution $y$.
To conclude, the standard practice is to compute a series of approximations to have a sharper estimation of $r$, for example employing several values for $N$ as $N_0, 2N_0, 4N_0, 8N_0,\ldots$, and then averaging $r$ over successive refinements with the logarithm formula above.
For time and space dependent problems, the estimation of error in time and space is often independent, as in $$ \|y - y_{N_x, N_t}\| \leq CN_x^{-r} + CN_t^{-q}, $$ where $N_x = 1 / \Delta_x$ and $N_t = T / \Delta_t$. In order to "see" the orders $r$ and $q$, one should apply the procedure described above first in space and then in time (or viceversa). In particular, one should first fix a big value for $N_x$ (good accuracy in time), and find $q$ by varying $N_t$, and then fix a small time step (big value of $N_t$) and apply the reverse procedure in space to find $r$.
Final note: for some time-dependent PDEs and numerical schemes, you have constraints in the choice of $\Delta_x$ and $\Delta_t$ in order to obtain a stable method (see CFL condition). Your search for orders of convergence should then be restrained to the "safe" values of $\Delta_t$ and $\Delta_x$.