Sorry - I kind of agree with Briggsy's Girl and Cass on this. There is a full three months between your birthday and your EDD. Your OP seems to be about the timing of having a baby shower - which is a fair enough question but you seem to be asking it because you are trying to time and dictate the type of presents that you receive? (Sorry if I've misunderstood you but that's how it reads to me).
No matter when you celebrate each event, people have the right to buy or not buy you presents as they wish and they also have the right to choose what they will give you and/or your baby.

I don't think you can even expect that people will automatically want to buy you or your baby a present - most will of course - but expecting it and being worried about whether people buy you "stingy" b'day presents because they've already spent money on the baby does sound a bit selfish TBH. Finances are a big issue for a lot of people at the moment with the global financial situation.

My birthday falls right before Christmas and I actually asked for baby stuff for my birthday and Christmas because I didn't want my family to feel they had to buy for me now and then again for baby only a couple of months later. Mostly I still got stuff for "me" though, with a few things for the baby. I was asked the other day by a close friend if I wanted a baby shower. I am 32 weeks now. I am seriously considering not having one because it just seems like an excuse to 'make' people buy things for the baby and that makes me feel guilty. I am really not sure what to do because I want to catch up with everyone before the baby arrives but living where I do, most of my close friends/family will be travelling at least two hours to attend and that feels like enough of a gift in itself. I have been trying to work out how to STOP people buying things for the baby without causing offence. I know this is unusual and is my issue and I'm not trying to put it on you as the 'right' way to be because lots of people really enjoy getting involved and giving gifts. I guess though I am coming from almost the opposite direction on this issue.

Look, at the end of the day, it's your birthday and your baby shower and you can do what you want. I don't think waiting until after your birthday is leaving it too late at all though - look at me!! If I have one at all, it will be at 36+ weeks. The only thing I would say would be to wait until your baby is at least viable - so towards the end of the second tri.