jmc

Let $p$ and $q$ be the prime roots. Then, we know that $m=p+q$ and $n=pq$. Since $m<20$, the primes $p$ and $q$ must both be less than $20$.

The primes less than $20$ are $2,$ $3,$ $5,$ $7,$ $11,$ $13,$ $17,$ $19.$ Now we list all possible pairs $(p,q)$ such that $p+q<20$, remembering to also include the cases in which $p=q$: $$\begin{array}{rl}& (2,2),(2,3),(2,5),(2,7),(2,11),(2,13),(2,17)\\ & (3,3),(3,5),(3,7),(3,11),(3,13)\\ & (5,5),(5,7),(5,11),(5,13)\\ & (7,7),(7,11)\end{array}$$There are $7+5+4+2=18$ pairs in total. Each pair produces a value for $n$, and furthermore, these values are all distinct, because every positive integer has a unique prime factorization. Therefore, there are $\boxed{18}$ possible values for $n$.