Childminders to my DS said exactly the same thing almost word by word when he was 6. I couldn't find any childminder willing to help as they all spoken to each-other. In my experience you don't get any luck with private providers. You need to find a publicly funded place. Talk to HV, sure start, find out about public after school care in your wider area (i.e. not just your village, but district, it might be some distance away). As View said, they are much better for children with SEN and disability.
Try to find some council run after school club, even for disabled children. Are there other schools in the area with after school care?Get your DS there. You may need to ask for a taxi to be arranged to pick him up. All council are supposed to have "local offer" on their websites, where there should be some offer of childcare for children with disabilities. Contact them and all people that can help you to get your DS there. Consider even moving DS to that school. Look at which local schools are good at meeting needs for ASD. Don't attach yourself to posh indicators of school 'goodness', they may not do anything tangibly useful for your DS if indeed he has ASD and ADHD. You need the school that will make SEN provisions, provide childcare and enable you to finish your degree.
Your DS sounds like ASD with possible ADHD. Write down your observations of behaviours and problems, keep a diary every day of what happens at school and at home. Give a print out to the GP and demand an urgent referral re ASD. Use the childminder refusal and the hampster incident to stress the urgency. Ask the GP to write to the Pediatrics again stressing urgency and new information, so they would prioritise him.
It is not correct that the school needs a diagnosis to start support. To the contrary, they need to start addressing difficulties they see regardless of diagnosis. See the Code of Practice for SEN, I am afraid you will have to become very familiar with it. Meet or write to the Senco show her the code of practice [literally hold it in your hands and open on the relevant page] and quote the relevant paragraph number X.XX. She needs to see that you are clued-up.. sadly. They may be arguing that they don't see any difficulties, but that is because due to funding they are trying to kick everything in the long grass they don't recognise what they see as sign of a SEN or disability. They may be mislabeling it as poor behaviour, being disruptive, being non academic, whatever. Your job is to learn all various ways ASD and ADHD can manifest itself and carefully catalogue, file every daily incident under the potential ASD/ADHD symptom, keep a daily diary. Talk about it to the school and ask to put support in place. Your DS's outbursts might be symptoms that his needs are not met, he might become more calm with the right understanding and provisions.
I would not defer uni in your place, as ADHD and ASD don't go away, it does not get any easier, every year new challenges emerge, so I would say the escape is in running forward,, IYSWIM. Try to find a suitable after school care that will work in long term and finish your degree.
I have 2 DC with SEN, ASD, ADHD. It is a struggle for life, it is draining and stressful, and at every turn there is a new challenge. So it is important to look after number 1, you. Do what is important to you, to feed your energy, emotions and sense of self. Your degree is clearly important for you, quite rightly. Don't second guess it. It will give you energy and confidence, and hopefully money. Just hang on and find an after school care that will accept your DS for the long term.

