Will cut-and-paste my post from Chat:
Although I am not a single parent, I can empathise with your situation greatly. My ds is almost 6yrs old, non-verbal, has severe learning difficulties, challenging behaviours, poor comprehension, doubly incontinent, blah blah.
Because your ds is 3yrs old, your first priority should be to find him a pre-school placement. There will be one that will take him; Social Services Children with Disabilities department may be able to point you in the right direction, or your local council will know where the nearest Children's Centre is. Your ds is entitled to at least 12.5hrs per week and as others have said, he may also be entitled to additional funding to provide him with a 1-1. When you get your ds into pre-school, this will give you some valuable and much-needed day-time respite, where you can relax a little, catch up with your own life and sort things out at home. Of equal importance though, is that your ds will start to receive professional input for his social, communication and behavioural skills.
Your next priority should be to plague SS. You have already contacted them and they have told you, 'no'. This is their default position but it is your job to have them change this 'no' to a 'yes', preferably by constant letter-writing, followed up by complaint letters (to which they have got to respond). Remember: phones bad, letters good. Keep a paper-trail so they cannot renege on anything that has been agreed. Have them come round to assess your ds and if needs be, to assess you as a carer too. Be prepared for a long fight to secure the respite care that you so patently need - they will NOT hand anything to you on a plate. You may be given Direct Payments or they may provide a worker to come and take your ds out for short periods. FWIW, my ds has just started to access a council-run respite centre (where he can occasionally stay overnight) but this has taken TWO YEARS to come to fruition.
I would add that the Statementing process for my ds started when he was about 3.5yrs old - this is another reason why your ds should be accessing pre-school. My ds attended a Children's Centre where he was regularly assessed by an O.T. a SALT and an Educational Psychologist. All their reports fed into my ds's eventual Statement.
I hope your relationship with your partner gets back on track but I would recommend telling your parents that you need support; perhaps they would be able to care for your ds every once in a while to allow you some free time?
If your ds is too difficult to take onto public transport, then you could make inquiries about getting him a McLarens Major buggy. Again, a referral was made from my ds's Children's Centre to our local Wheelchair Services who then provided us with a Major - which we use even now he is almost 6yrs old.
Are there any parent support groups in your local area? It may be worth contacting your local Special School to see if run any groups.
I hope things improve for you soon. It is hard but it is not your fault and you do need support and so does your ds. Good luck and HTH.