The Hull-White short rate model (or any other short rate model) describes the short rate dynamics $dr(t)$ as well as provide the analytical solution of the zero coupon bond $P(t, T)$:
$$
P(t, T) = E_t^Q \left[ \exp \left( - \int_t^T r(s) ds \right) \right] = \exp(A(t, T) - B(t, T) \cdot r(t))
$$
Depending on the notation you are using, the zero coupon bonds can differ with the previous expression, however, the following still applies. Once you find the solution for $A(t, T)$ and $B(t, T)$ by means of a system of ordinary differential equations (Riccati System of ODEs), you can compute, for each simulation path, the zero coupon bond at any $(t, T)$.
Consider a vanilla swap described by the tenor structure $T$ such that $0 \leq T_1 < T_2 < \dots < T_N$, its swap rate is given by:
$$
S(t) = \frac{P(t, T_1) - P(t, T_N)}{\sum_{n=1}^{N} \tau_n \cdot P(t, T_{n+1})} \quad \text{with } t < T_1,
$$
then you can compute, for each path, its swap rate.