RAID teaches that in any undemanded "Reg out of mouth" scenario, step 1 is to get breathing gas. This will almost always be by taking the alternate and placing it in your mouth.
Step 2 is assessing what happened. It may be a normal loss (mouthpiece came off, kicked out of mouth, etc) or you may find a diver with big eyes and no gas at the other end, in which case this is now an air sharing ascent.
In terms of this, my go to recovery is usually just to look to my right, I can almost always see it hanging there. Since there is no rush, I can then ask my buddy if I can't see it, sweep for it, follow hoses etc etc
I teach my students all these methods. I also teach them in standard rec setup where necessary, and how to make a pull away bungee for when they don't have access to suitable "primary donate" equipment.
I used to HATE teaching the reg recovery. Watching the student blowing out a little stream of bubbles while they sweep and KNOWING that if they miss the first sweep then its likely going to be a bolter, always stressed me out ridiculously. Watching the anxiety set in as they realise they are running out of time was terrible for everyone involved.