I have found that sometimes I have use Scrubbing Bubbles for some of the harder stuff left on them
I agree with the many comments here that warm water and dishwashing liquid should be enough for the razor, provided there is no significant rust, which there probably isn't, especially with a nickel-plated brass razor. The steel blades, on the other hand, are another matter. It always puzzles me how some here are almost fanatical about "sterilizing" vintage razors, when it is the blades that have the most contact with the skin, sometimes even penetrating it and drawing blood. I doubt there is much chance of infection, but I would examine any old blade very carefully to make sure there is absolutely no rust before using it. Better yet, keep the old blades for the classy vintage look but use new ones.