If you can afford it, a solicitor with experience of applying for a licence back early is worth it.
If you have to go it alone, here is what you do:
Write a letter to the court that disqualified you, asking for a hearing to make an application for the restoration of your ban.
Set out the date you were banned and the length of the ban.
Then set out the reasons why you want your licence back early.
Start off by saying what you have done to alter your drinking habits since you were banned. Have you had any involvement from your GP or any other medical help? detail them. That helps then assess the risk you pose now.
Attach any character references that you can get (2 is enough) to say something about your character. One should be from work and hopefully include any details of how you will be of more use to them if you have your licence back now. This could be about being able top do extra late shifts or visit different branches that cannot be done on public transport.
Then detail the impact that becoming a father will have. (No good talking about the actual birth, because it will take a couple of months to get a hearing and another 2-3 months to go through the DVLA medical process)
Then anything that will help others if you get your licence back early.
What is not good is saying how hard it is on YOU.... after all you were banned (presumably) for 44 months because of a second conviction in 10 years and a high reading or a bad accident this last time, because the usual is 36 months. It is supposed to be hard on you!
Sent that to the court, who will notify the Police and CPS of your application and give them the opportunity to make representations as to why you should not get your licence back early, or say that they have no objections. (even if there are no objections it is still at the discretion of the Magistrates)
The reason for putting in as much information as you can is that if there are compelling reasons, it makes it less likely that the police or CPS will object. If you just write and say "I have served half or my ban, 2 years have elapsed, I want my licence back." you can be sure that they WILL turn up to object!
Note that it doesn't mean that what is in the letter is the only things that you can put to the magistrates, you can elaborate on the day as well.
Hope that helps.....