  Atomic clocks based on an Al$^+$ ion sympathetically cooled by a laser-cooled
alkaline-earth ion have achieved unprecedented accuracy. Here, we investigate
theoretically interactions and charge transfer dynamics of an Al$^+$ ion
immersed in an ultracold gas of Rb and Sr atoms. We calculate potential energy
curves and transition electric dipole moments for the (Al+Rb)$^+$ and
(Al+Sr)$^+$ ion-atom systems using coupled cluster and multireference
configuration interaction methods with scalar relativistic effects included
within the small-core energy-consistent pseudopotentials in Rb and Sr atoms.
The long-range interaction coefficients are also reported. We use the
electronic structure data to investigate cold collisions and charge transfer
dynamics. Scattering of an Al$^+$ ion with alkali-metal or alkaline-earth-metal
atom is governed by one potential energy curve whereas charge transfer can lead
to several electronic states mixed by the relativistic spin-orbit coupling. We
examine the branching ratios resulting from the interplay of the short- and
long-range effects, as well as the prospects for the laser-field control and
formation of molecular ions. We propose to employ the atomic clock transition
in an Al$^+$ ion to monitor ion-atom scattering dynamics via quantum logic
spectroscopy. The presented results pave the way for the application of atomic
ions other than alkali-metal and alkaline-earth-metal ones in the field of cold
hybrid ion-atom experiments.
