A lot of this depends on the citizenship(s) of you and your wife. In general, being born in the UK is not sufficient to become a UK citizen. Assuming, the newborn is not a UK (or EU) citizen, it is not clear if you technically need to apply for a visa for your newborn: https://www.gov.uk/tier-2-general/family-members
If you have children while you’re in the UK, you can apply for permission for them to stay.
You must do this if you want to travel in and out of the UK with your child.
If you plan on never leaving the UK (even for a holiday), potentially the newborn baby might be able to get away without a visa.
You will likely have to get a passport for the baby prior to applying for the dependent visa. When I completed the application for my son, the process was a bit tricky since it is not clear if it is the sponsor or the dependent who is applying. We used a immigration lawyer to help with the paperwork. Further, the question regarding "how did you enter the UK" is a bit tricky to answer since your only choices are air, sea, channel tunnel, and overland. My wife would not let me choose channel tunnel, despite my arguments that it was the most accurate.
You will likely have to pay the NHS surcharge for the new baby, but everyone should have access to the NHS. It is important that you and your wife have valid visas and are qualified for the NHS at the time of birth so you do not get charged.