Can a runner advance on a check swing appeal call?

In your situation, the runner on first gets to go to second because it is a walk. The first base ump shouldn't be making calls when the home plate ump already called ball 4. (The catcher's appeal means nothing. Only the home plate ump's appeal actually matters.) The home plate ump call takes precedence over any call, so the runner on first base would get to advance to second because of the walk.