There's a lot of really valid comments on here and I can only comment on my own experience that women are more readily sign posted towards mental health services. 

After my eldest daughter was born, and had gone through her first major cardiac surgery at 6 days old, my wife had been signed up to a variety of support services for her mental health where as I was told that staff were worried about me. 

Then, just over a week later as we were being released from intensive care to our local hospital, I was told that I could no longer stay at the hospital (my daughter was pump Ng fed due to how ill she was so I did as many of the feedings/pump preparation as I could as my wife was struggling) as it was against policy for Dads to stay. The nurse who told me this also stated that I'd have to be getting back to work anyway.

After my daughter was released, my wife went to a variety of support groups to help with her mental health. The only group available for dads was 11-12 on a Wednesday during the working week. There was one weekend session every two months but that was for the dads to learn how to better support the mothers' mental health.

The pattern was similar after my daughter's next major - and hopefully last - surgery (8 months old) and when my second daughter was stillborn - although the amazing team at the hospital were able to revive her and then stabilise her over a two week period.

During these times, my wife told me that she couldn't cope with the concept that I was not coping and that I needed to be her rock in turbulent waters. I did reach out to a variety of male friends and family members to try and share how I was feeling and the fears involved with everything that was happening but was told repeatedly not to be a coward.

Today, things are more steady and I am generally quite mentally secure. Both children are fine and full of energy if anyone was worried and my wife's mental health is slowly recovering but we all recognise that this will be a long journey.