jmc

Note that $60$ is divisible by $3$, but $20$ is not divisible by $3$. Therefore, if $\mathrm{lcm}[\nu ,20]=60$, then $\nu$ must be divisible by 3 and we may write $\nu =3n$ (where $n$ is a positive integer).

Thus we have $\mathrm{lcm}[3n,20]=60$, and since the $3n$ contributes the factor of $3$ to $\mathrm{lcm}[3n,20]$, it follows that $\mathrm{lcm}[n,20]=\frac{60}{3}=20$. This is true if and only if $n$ is a divisor of $20$. Therefore, the possible values of $\nu$ are $3$ times the positive divisors of $20$: $$\nu =3,6,12,15,30,\text{or}60.$$The sum of these values is $\boxed{126}$.