I disagree strongly with the idea that a 13 year old shouldn't be left at home alone, or that people should live in fear of Social Services if they choose to do so. The vast majority of schooled 13 year olds go to and from school alone and it is extremely common for them to be in the house unattended for several hours at a time. A parent would only be convicted of neglect if the court finds that it was unreasonable to leave that particular child under the circumstances, that a sensible person would have anticipated that he could well have a serious accident. Anyone can have an accident, even an adult.
With only a few exceptions, all of my 13yo's friends (schooled and HE) are regularly left alone for hours at a time. The exceptions are where there are unusual circumstances, such as the young person having shown particular difficulty coping safely with everyday challenges such as remembering to switch off the grill or being sensible about dealing with people who come to the door.
Having said that, in the long run being alone in the house full-time could be isolating and I think it would be wise to look for ways to get him out of the house and keep him active, and find people to keep him company. In view of your son's extreme distress, if I were you, KerryLou, I'd pull him out of school straight away and then begin to put your mind (and his) to work on how you can keep him happy and prevent loneliness.
You could, for example, have your son spend part of the week with a childminder, relative or family friend. Perhaps you could consider changing your job or your working hours so that you will be with him more. Can he get out under his own steam to the swimming pool, library or shops? Are there volunteer opportunities in the area? Would he like to walk the neighbours' dogs or work on their gardens? Once he's begun to recover somewhat from the bullying, maybe he will want to join an after-school club of some sort.
You'll find that with individual work and no distractions, your son is able to get through his academic work quickly and he will need things to keep him occupied. Unless he loves to be alone, sitting in the house alone all day every day could soon begin to feel like house arrest. It could be OK as a short-term solution to the current crisis - chances are it would still be better than school for him right now - but eventually you'll need to make arrangements which can meet his social and emotional needs more completely.