In branching diffusions with immigration (BDI), particles travel on independent diffusion paths in $\mathbb{R}^d$, branch at position-dependent rates and leave offspring -- randomly scattered around the parent's death position -- according to position-dependent laws. We specify a set of conditions which grants ergodicity such that the invariant occupation measure is of finite total mass and admits a continuous Lebesgue density. Under discrete-time observation, BDI configurations being recorded at discrete times $iΔ$ only, $i\in\mathbb{N}_0$, we lose information about particle identities between successive observation times. We present a reconstruction algorithm which in a high-frequency setting (asymptotics $Δ\downarrow 0$) allows to reconstruct correctly a sufficiently large proportion of particle identities, and thus allows to recover $Δ$-increments of unobserved diffusion paths on which particles are travelling. Picking some few well-chosen observations we fill regression schemes which, on cubes $A$ where the invariant occupation density is strictly positive, allow to estimate the diffusion coefficient of the one-particle motion at nonparametric rates.