We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of Kleiner and Lott [Acta Math. 219 (2017), pp. 65–134]. Results of Bamler-Kleiner [Acta Math. 228 (2022), pp. 1–215] and Haslhofer [Proc. Amer. Math. Soc. 150 (2022), pp. 5433–5437] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared.