Why do the cations of Group 1 increase in radius with increasing proton number?
Because the number of electrons increase down group 1.
Generally atomic radius always increases down groups and this is true for ions as well. Down any group the number of electrons surrounding the atom increases and this also means that the number of shells of electrons surrounding the atom also increases. Therefore, down the groups the atomic radius increases.
Another factor can also be taken into consideration. This is the repulsion that occurs between shells of electrons. As the number of shells increase so does the number of negative charges surrounding the positive nucleus. Thus the amount of repulsion between shells also increases as the number of electrons and shells increase.