I did a few times of BW during my pregnancy. Your skin will be more sensitive during pregnancy (as with before your period), so it will be more painful than usual.
Won't be itchy after that, but when the hairs start to grow out it may be itchy because of your panties rubbing the area.
I've been to Honeypot and like the service there. Their waxing is not as painful as at Strip.
I did my BW around week 35-36, but my hairs grew back so quickly that when I delivered at week 40, they still had to shave for me. So maybe you might wanna get it done around week 38?
Do note that if you get your BW at these places, they will ask you to sign a consent form stating that they will not be liable in case anything happens to you. As far as I'm concerned, having a BW did not affect my baby much. Just dont't tense up and relax as much as possible. Good time to practice your breathing techniques too!