Oh you poor love, you have every single ounce of my sympathy. I had two terrible sleepers (my youngest woke up 8-9 times a night until a week before his third birthday!). It is maddening. It drove me straight into illness. It is just so hard, beyond description really.
The only thing that worked for me was DH giving me a night of full sleep once in a while. We just said to each other, "We're in the trenches here," and gave each other lots of room to be miserable and exhausted. I think just owning the awfulness of being permanently and woefully underslept gave us strength to carry on. We agreed, no pestering for sex, no asking anything of each other. Just hugs and respite once in a blue moon. I felt like I could somewhat manage life on little to no sleep once DH and I gave each other permission to just get through it, one day at a time, without any pressure or obligation to fulfill needs. My mental health really improved.
We were also really clear with people. If I cancelled on friends due to utter exhaustion, they just either understood or they didn't. I couldn't really care about letting people down. I was too bloody tired to care. Fortunately, I have really loving friends who were totally understanding.
You are letting nobody down. This isn't a challenge or a performance. You don't need to do anything else other than get through it. That's all you need to do. 
What I did not have to deal with is hysteria. That is utterly soul destroying. Here in London, my youngest was referred to the paediatric sleep clinic at the Evelina children's hospital. It's part of paediatric neurology. Anyway, by the time his appointment came up, he was sleeping through the night. So I never got to use the service but did have a couple of good phone consultations beforehand.
www.evelinalondon.nhs.uk/our-services/hospital/sleep-medicine-department/referrals.aspx
If I were you, I would push my GP hard for a referral to your sleep medicine clinic. You can't keep going. And your little one's development will be inhibited (he will totally catch up! It will be a temporary thing, but lack of sleep will certainly slow down development. DC3 was a different child once he slept. His development just leaped like mad once he started sleeping through).
The Evelina Hospital's sleep clinic also has an app called Kids Sleep Dr which is really worth installing. It won't solve the problem, but it will be like a diary. You'll understand your child's sleep pattern better and perhaps notice what is impacting your child's sleep. Most sleep clinics will want you to keep a diary anyway and this gives you a head start. You can attend the clinic armed with information on your child's sleeping habits/disruptions, etc which will help move things along a little bit faster.