You really won't remember to much about the recovery room, and usually feel fairly well with all the drugs you are on, so I'm sure you'll be fine. The recovery nurses are absolutely beautiful, and honestly you won't worry at all about no-one familiar being there, its more about getting your bearings, and realising the surgery is finished.
Best of luck, and a great recovery to you.
Usually hospitals don't let people in recovery unless you work there all for infection control reasons. You will be fine, i was worried about my operation i had in november to have a dermoid cyst removed, when i woke up i couldn't believe that i had actually had the operation done as it only felt like 5 minutes that they had taken me into operating theatre. The nurses in recovery are always lovely they do this job everyday so you will be in good hands, i was chatting away about pasty after i woke up, so you will be fine, i don't remember them taking me back to my room!
I do not know what the policies are at different hospitals in different countries, but it seems unlikely that anyone other than doctors and nurses are in the recovery area with patients.
The recovery area is where the medical professionals watch and wait for you to awaken, then when they are confident you are coming out of anesthesia properly, they take you to your room. After that you can be visited by friends and family. Trust me, you will not be alone in recovery. They are quite attentive to patients that just got out of surgery. I was not in recovery long after I first opened my eyes.
You will be really foggy when you first awaken. You won't really know or care who is there, just that you are "awake" and that the surgery is behind you. I barely remember the trip to my room, but my husband appeared there almost immediately. I am sure whoever accompanies you to surgery will do the same.