You will face difficulties with your plan.
Dad of an EU citizen will mean nothing, so you can forget about it. Also keep in mind, that depending upon the hospital and how clued up they are about the rules, if your wife delivers in UK, and hospital is a bit clued up, you might face a big bill. If they do bill you, and you live in UK, then be assured they will chase you all the way to recover money.
You whole status in UK will depend entirely on your wife exercising treaty rights. Your wife being pregnant cannot work, so she can't exercise treaty rights. The way around it is, that if your wife has cash savings, the she can obtain residence card based on being "self sufficient". Both of you will need comprehensive sickness insurance to succeed in this category. After this, if you can start work, then your wife can still be regarded as self sufficient because you are working, and you get your rights because she gets her rights. Call it ping pong. Keep in mind, your wife has to continuously exercise treaty rights for 5 years for both of you to gain permanent residence in UK.
regards