Electric field due to a hydrogen atom

Especially the hydrogen atom, with a proton in the nucleus and an electron revolving acting as a dipole

This is a problematic way of understanding the hydrogen atom ─ it basically tries to insist on treating it within classical mechanics, and this is doomed to fail. Instead, the hydrogen atom must be treated within quantum mechanics. This introduces a bunch of subtleties, but as an initial answer, you can think of the hydrogen atom as a stationary proton surrounded by a spherical 'probability cloud' in which the electron can be found (if and when you go looking for it).

Since this probability cloud is spherically symmetric, its electric field will completely cancel out that of the proton, and the total electric field produced by the atom will vanish. This is generically the case for all atoms sitting unperturbed in vacuum.

That said, though, it is possible to polarize the atom if it is placed in an external electric field, which will displace the center of the electrons' probability cloud, and that will produce a small dipolar electric field. Furthermore, this can also be in response to the internal charges of a different neutral atom, which will produce a van der Waals force between them. So take that zero-field statement with a grain of salt!