Find one-variable recursive formula for running time of Karatsuba multiplication
The recurrence with two variables $n,m$ is as what you have obtained:

$$T(n, m) = 3T(n/2, m/2) + O(\max(n,m))$$

Assume $n = \Omega(m)$. We have

$$T(n, m) = 3T(n/2, m/2) + O(n)$$

Although the master theorem cannot be applied directly, if you expand this recurrence, then you will eventually identify the patterns and the result:

$$T(n, m) = 3T(n/2, m/2) + O(n) = \cdots = \sum_{k=?}^{k=?} (3/2)^{k} \cdot O(n) + c \cdot T(n/m,1) = \cdots$$
Please find out $c$ and the range of $k$, and fill the gaps.



There is a method to reduce the two variables $n,m$ recurrence to a single one:


  Assume $n = \max(n,m)$ and pre-append $n-m$ $0$'s to the integer with less bits. 


The problem is this will yield a different time complexity from that given above. This can be seen from an extreme example: $1$-bit $\times$ $O(n)$-bit.