What sidearms did the crews for M1919 mg, M2 hmg, mortar and recoilless gun use?

Well, I assume we're talking WW2. (All of those weapons were used in Korea, Viet Nam and in many cases are still in use.)

The members of the weapons and heavy weapons platoons would have been armed with the M1 Garand and M1 Carbine, or perhaps the Thompson sub-machine gun (for NCOs). They were not issued pistols, although they may have scrounged one up.

The actual gunners of the M1919 and perhaps Bazooka too, may have only had a Model 1911 45. auto pistol for personal defense, as their 'primary' weapon was the gun itself.

The M2 MG and the 60 & 81 mm mortars were too heavy to be considered 'personal' weapons, and couldn't be used for personal defense, so their gunners would have been armed like everyone else.

Officers would have .45 pistols, and perhaps a rifle, as well.

At least that's the way I understood it...

Machine gun teams; gunner had a M1911 .45 pistol. The assistant gunner originally had a M1 rifle, later changed to a M1 carbine.
Mortar crews were supposed to have a mix of carbines and rifles. Dedicated crew the carbines and the designated ammo carriers, rifles. Section leaders were authorized a sidearm.

Recoiless rifles and bazooka teams were supposed to have carbines and the section leader a sidearm. Ammo carriers had rifles.

In Korea replace the M1 carbines with M2 carbines. The M14 rifle replaced both rifles and carbines. Yes, there were M1919 teams after the adaptation of the M14 for a short time before the M60 lmg came along. Really an anachronism to see guys with M14 lugging along an M1919. The M16 was used in SE Asia in lieu of the M14, the M14 being reserved for USAEUR.

And the National Guard retained M1s into the early 1970s.

About what time did the mg assistant gunner change from M1 garand to M1 carbine?

In an actual unit with casualties and replacements it would be come as you are. It could be anybody grabbed from anywhere to service the machine gun and what the TO&E said didn't matter.

The changes filtered down in 1943 but were never universal. Some guys hated the carbine and held on to their M1. Others ditched their rifle at first opportunity and got a carbine.

But carbines weren't widespread until 1943.