It is a common misunderstanding that the double slit experiment is because you shoot electrons at the same time and they interfere.
In the double slit experiment, in reality, you shoot one single electron at a time. That electron acts as a wave on the fly, and its partial waves pass both slits at the same time, and these partial waves interfere with each other. The interference between the partial waves can be constructive, that is where you see a dot on the screen.
The interference can be destructive, that is where you see the dark areas on the screen.
The wavefunction of the electrons is the probability distribution of finding the electron at a certain position in all of space.
Now when the constructive interference happens, we use the phrase, the wavefunction of the electron collapses. In reality, nothing collapses, but the one single electron we just shot will make a dot in the screen, its wavefunction will reduce to a certain eigenvalue, that is a position on the screen.
To see an interference pattern on the screen, you need to shoot a lot of electrons after each other.
You are asking if you shoot an electron, then a proton, then an electron, then a proton, then will you see a pattern on the screen?
If you set up the electron pump (that shoots the electrons) so that each electron will have the same characteristics, then you will see a pattern for the electrons.
If you set up the proton pump (that shoots the protons) so that each proton will have the same characteristics, then you will see a pattern for the protons.
What you will see, is two patterns over each other on the screen. One pattern will be for the electrons, and another for the protons. But this will not mean that there is any interference between the electrons or the protons.
You are asking whether it is possible to shoot the protons and the electrons so from the same source that both type of particles will add to the same interference pattern and you could just see one interference pattern. This is not possible, since the wavefunction of the electrons and protons would have to be the same, and their quantum characteristics would have to be the same (it is not, electrons and protons are different quantum systems, with different quantum characteristics).