One nice piece of evidence comes from Supernova 1987a, which was special because it was not very far away. Theory predicts that such a supernova would create about 0.1 solar masses of nickel-56, which is radioactive. Nickel-56 decays with a half-life of 6.1 days into cobalt-56, which in turn decays with a half-life of 77.1 days. Both kinds of decay give off very distinctive gamma rays. Analysis of the gamma rays from SN1987a showed mostly cobalt-56, exactly as predicted. And, the amount of those gamma rays died away with exactly the half-life of cobalt-56. For more details, read:
The Compton Gamma Ray Observatory, Neil Gehrels et al, Scientific American, December 1993, pp.68-77
SN1987a Light Curves, P. Whitelock et al., in Proceedings of the Tenth Santa Cruz Workshop in Astronomy and Astrophysics, Springer- Verlag, 1991.
Since SN1987a was 170,000 light years away we were seeing light generated 170,000 years ago. This means that radioactive decay ran at the same speed 170,000 years ago as it does now.