Hi John and welcome to the forum. Sorry to hear of your situation which is not dissimilar to my own and many others whom contribute to this site. Based on my experience to date this is my advice to you.
I would expect that your ex partners hormones will be running wild at the moment having recently given birth and she may be somewhat overwhelmed by the whole experience of now having a baby to look after. Emotions will be running high for you both and as we are all aware in these situations things get said as an emotional response that would not be said under normal circumstances.
Your child is entitled to a relationship with you and your family and this is something that the courts will enforce if necessary. However, before going down this route I would first exhaust all other options. I would take the approach of writing a non-confrontational letter to your ex stating how much you love your child and how much you want to play a big part in their life, provide emotional and financial support and that you're in for the long term. State that you hope to agree regular structured contact that is in line with your child's emotional needs. Don't make any threats about court etc as this will only make the situation worse. Before sending the letter sleep on it, review it and ask a friend or family member to review the letter objectively to ensure it is non-confrontational and is written in the correct tone. Review the letter and amend as necessary. Sleep on it once again before delivering. Another option is to send a letter to a family member whom you've had a good relationship or feel would be open what you are saying and explain the position to them, again don't make it confrontational, keep to the subject of contact and don't make comments about the ex. See what response you receive.
You could also send occasional texts to your ex partner asking how your child is, again showing your interest as a father. Keep the subject of the texts to asking about your child. Don't expect a response and don't rise to any off-hand responses you may receive.
In terms of contact arrangements I think that given the age of your child you can only reasonably expect that contact will take place at the home of your ex given that your child will be wholly dependent on your ex at the moment and also that the ex may still be recovering from the birth. I've been in the situation whereby I had to visit my daughter at her grandmothers home for six months and was treated like dirt however; I turned up on time, every time, was polite and spent the duration of my contact with my daughter. Upon leaving I was polite and thanked her for hosting the contact. Don't get into any discussions about contact, maintenance etc etc. by doing so you are showing that your interest is solely for your child and that you're in it for the long haul, it will also show that no matter how hard they think they're going to make it for you you won't be put off. Unfortunately, you have to suck that one up and get on with it...smile, even if through gritted teeth and kill them with kindness.
Keep a diary of all contact, what took place, when it took place, the duration, what you did etc etc. Also record where contact has been arranged and withdrawn if this occurs. Keep receipts for everything you buy and make any maintenance payments via bank transfer clearly labelled as 'child maintenance' In short, record everything and keep text messages (back them up if possible). This will be useful if you have to go down the legal route.
I'd go with any contact you are offered, if it's not convenient say so and try and arrange a time that's suitable for all of you. Take a hands on approach to doing nappies, bathing, feeding etc. It will all stand you in good stead with your ex and do you no harm when you want to increase the arrangements if your ex has confidence in your ability to take care of your child. Again document what you've done.
After a while ask your ex to increase the frequency and duration of your contact and introduce additional activities for example taking your child out for a stroll in the pram. If this is turned down, ask why.
If after a period of time you feel that you're not getting reasonable contact suggest mediation to your ex. If you're on a low income financial support for mediation is available. You'll find links in this site to the info. If you feel that court is the only option and money is tight you can always represent yourself. This will cost £200 for submission of the contact forms. Again there is plenty of information available on this site to help you.
In terms of how you are feeling, try and relax, tell yourself that you're in this for the long term and although things may seem overwhelming at the moment things will get better in the long term.
I wish you well.
LD