Very briefly (less than an nanosecond (10-9 s), and probably on the order of 10-11 s). [I'll try to find an open source reference] Note that a beta particle is an electron and it will have a short range in the matter. It may collide with other electrons in the original atom, but it will interact with other electrons in neighboring atoms. The range in the material depends on the energy of the beta particle and type of material (electron density and binding energies of the atomic electrons). A positive ion will 'steal' an electron from a neighboring atom, which sets off a cascade of electrons jumping from other atoms until the original beta electron finds an ion and recombines.
When a positron (sometimes called a positive beta particle) is emitted, the atomic charge goes from Z to Z-1 and an atomic electron is emitted. The positron slows down through collisions until it combines with a negative electron and the pair annihilates into two photons of ~0.511 MeV. That is the basis of a PET scan. The atom that loses an electron then steals an electron from one of its neighbors and that starts a cascade of electron stealing until atom absorbs the free electron from the atom that emitted the positron.
In the case of the alpha particle, the atom number decreases by 2, i.e., Z -> Z-2, and so two electrons are lost. The alpha particle will slow down in the vicinity of the atom from which it came and become a helium atom, while the emitting atom will lose two electrons. Those free electrons will migrate to an available ion and recombine.
