About 2 months.
It has to heal, which is short time for the normal micro-surgery, you have to go to rehab to lay the good foundation for the processes for redeveloping the muscles, then you actually have to build up the bowling specific muscles.
Muscles not being used deteriorate rapidly.

Mine was relatively minor and I was younger, 10 years ago.
It depends on how serious, detailed, complex the surgery is and how good your muscles are.
The younger and the stronger you are (without being muscle bound), the quicker it will/can go.

I found that you and the therapist (if he/she is good) are the best judges of how soon you can return. The orthopedist will let you know the basis for the judgement.

Like Charlest said it depends on how much needs to be done.

I had a torn labrum and partially torn rotator cuff fixed August 2015. I was back bowling with a 13lb ball 6-7 weeks later and back to my regular 15lb in about 10-11 weeks.

My situation was also a little unique as my orthopedic doctor didnt actually want me to do therapy, just rest. The muscles in my shoulder were in good health except for the 1 partially torn.

I think doctors don't always believe in therapy, which seems odd to me. (Some do; some don't.)
I just tore my hamstring; the doctor thought rest should do it and I'd be back in a short time. He gave me a script for therapy but said he didn't think it was necessary.
AT first I didn't go and just did some exercises he suggested. About a week later, I figured, what the heck, I got the prescription. If it doesn't help, I can always stop. The therapist was great; he was much more detailed in his analysis than the orthopedist, even though even he thought I should be back bowling very soon. He found which muscle sections of the couple of hamstring muscles (it's not just one) were damaged. He focused on them without ignoring the rest of the injury site. After 3 weeks with the therapist, I'm back bowling and with zero pain, after 8 weeks of not bowling. The therapist made a world of difference in my confidence and in my physical well-being. Get a good one and do yourself a favor!

Being your left shoulder is going to shorten the time, but I'd still estimate 6 weeks anyway.

Mine was a mess in 2014.  My bicep was torn, as well as the rotator cuff and they had to carve out some bone on the acromiom.  The doc actually drilled a hole in my upper arm and attached my bicep there, rather than in the shoulder joint.  (My right bicep is slightly lower than the left side, but strong now.)

Due to all the damage I really needed 4 months before bowling carefully, and 6-9 months until I felt confident in the repair.  I worked very hard at therapy the whole time because I really missed bowling.  Rotator cuff therapy is slow and tedious, but will make or break your success.  I remember doing tons of stretching and range of motion excercises, as well as very light weights when I made enough progress.

Hopefully you're young enough that your body will respond well for you.  Best of luck from all of us here at BR.

My dr. Said it was bicep tendonitis.  Gave me some steroids and if it's not better by next week gotta get a shot.  We shall see what happens.  League in Monday night but may still rest it.

Ya I am gonna still go and see my ortho just to be on the safe side.  Is there anything you can wear to help with bicep tendonitis when you bowl.