Please can I check, when you say the milk won't come through, what exactly is happening? E.g. do you get any milk at all when you hand express or pump? Or does the engorgement go down and your breast(s) start to soften when you feed your baby?
The engorgement means your breasts are producing milk which is a good sign. However, you'll want/need to work on relieving that engorgement for your comfort, to avoid mastitis, and to stimulate milk production (supply is all about demand... you won't produce milk unless it's removed). You could try massage along with the hot compresses just before a feed or pumping session (and if you notice any lumpy areas, focus on them and massage towards your nipple so that you relieve the blocked ducts).
Don't worry too much about what you produce when pumping or hand expressing... babies are much more efficient than a breast pump. The key thing is to put your baby to your breast as often as you can - I'd recommend lots of skin to skin and opportunity to feed. You can judge the input by the output (i.e. wet nappies).
Have you had any advice from the midwife about pumping/feeding? A common technique to boost supply is to feed your baby and then pump immediately afterwards. This works well (even if you don't produce loads of milk when pumping, it helps to stimulate milk production), however it is labour intensive at this newborn stage and you'll want to balance doing that with sleeping and eating enough, so it's worth asking someone else to sterilise/prepare your equipment while you rest/eat, etc. Your job should ideally be to rest (when you can) recover and feed your baby at the moment.
Are you still using formula? There's no issue with that in itself, but it's worth saying that giving formula can cause supply issues, because every bottle you give equates to a feed that you haven't stimulated your breasts to make (so it can become a vicious circle of worrying your baby isn't getting enough breast milk, giving formula to compensate, then your supply diminishing because baby isn't stimulating your breasts to make more milk).
It's early days and breastfeeding takes a while to establish. Go easy on yourself (especially while your hormones are raging!) and remember that you're doing a great job (especially following such a traumatic birth experience). 