How would an atom behave (in terms of atomic particles) if the electrons were much heavier than the nucleus? In other words, does the mass of electrically charged particles affect the electromagnetic forces that operate between them ?
Is the electrons revolving around the nucleus simply a consequence of the electro-weak and electromagnetic forces ?
In this case would the atom behave the same if the masses of the nucleus and the electron were inverted (say in a hydrogen atom)?
 A: The main force keeping the electrons around the nucleus is the electromagnetic one. Electrons do interact gravitationally and weakly but those are very much weaker forces.
In principle if the masses were exactly inverted, it would just change the definition of positive and negative charge, which is arbitrary. Generally changes in mass affect the orbitals.
The Bohr atom is useful in getting an intuition:

Please look at the link for the definition of the variables but it is easy to see the mass of the electron in the denominator. The larger the mass the smaller the radius.
There exist muonic atoms, i.e. where an electron has temporarily been replaced by a muon and the orbitals are much smaller and tighter.

Since a muon is more massive than an electron, the Bohr orbits are closer to the nucleus in a muonic atom than in an ordinary atom, and corrections due to quantum electrodynamics are more important. Study of muonic atoms' energy levels as well as transition rates from excited states to the ground state therefore provide experimental tests of quantum electrodynamics.
Muon-catalyzed fusion is a technical application of muonic atoms.

