Weyl semimetals have nodes in their electronic structure at which electrons attain a definite chirality. Due to the chiral anomaly, the nonconservation of charges with given chirality, the axion term appears in their effective electromagnetic action. We determine how this affects the properties of time-reversal invariant Weyl superconductors (SCs) in the London regime. For type II SCs the axion coupling generates magnetic B fields transverse to vortices, which become unstable at a critical coupling so that a transition into type I SC ensues. In this regime an applied B field not only decays inside the SC within the London penetration depth, but the axion coupling generates an additional perpendicular field. Consequently, when penetrating into the bulk the B field starts to steadily rotate away from the applied field. At a critical coupling the screening of the magnetic field breaks down. The novel chiral superconducting state that emerges has a periodically divergent susceptibility that separates onsets of chiral Meissner regimes. The chiral anomaly thus leaves very crisp experimental signatures in structurally chiral Weyl SCs with an axion response.