When an alpha particle is emitted, two protons and two neutrons leave the nucleus but the electrons remain the same in number. Why does the atom remain uncharged although it appears it should have a net charge of $-2e$?
It's not true that the atom is electrically neutral afterward. If you have a single atom, isolated in a vacuum, and all that is emitted is an alpha particle, then as you say, it has a net charge of -2e. In reality, alpha decay is a violent process that is likely to knock out some electrons as well. Furthermore, if the atom is in a solid, then electrons are being shared and exchanged all the time. Eventually, this may result in both the alpha and the decay-product atom becoming neutral again.
A decay destroys the electromagnetic wave function of the atom, the one that generates the energy levels which keep the electrons bound to it.
The new nucleus, after the deacy, will have a new potential whose solutions will have binding levels for n-2 electrons. The two extra will be left in the lattice ( or in the gas) free to join up in the energy levels of the potential of an alpha particle passing close to them. If the alphas leave the bulk matter, then that bulk will be negatively charged.
You're right, directly after the emission it will have charge -2e if it was neutral before, i.e. be an ion. But within a gas or liquid electrons are very easily exchanged and ejected. This is of course at a much lower energy energy scale than the nuclear emission and therefore less noticeable.
So the atom is left with two more electrons than protons and therefore has a negative net charge. This repels the two electrons, so they leave, and fast. And now the atom is neutral.