@GGee123 ACAS sets out the process for making contractual changes. https://www.acas.org.uk/changing-an-employment-contract/employer-responsibilities/proposing-employment-contract-changes
What strikes me is you are not being honest with yourself about what hours are actually needed to do your job. You are also confusing the number of hours you are contracted to work vs actual hours you are working.
From what you have said your Director/employer could make a fairly strong business case your job can’t be done in your contracted hours.
Your Director/work may believe in presenteeism, if that is a case you need to talk to your TU and other part timers as volume of numbers can make a difference if they are trying to get all part timers to work full time.
However, they may becoming at this from a duty of care and concern that legally they are not paying you fairly. Plus you are making a full time job look part time.
I can understand your concern about not wanting to be made ill again. In my experience managers and employees can be very limited when thinking about flexible working. Creative approaches can and do meet business needs as well as personal needs.
As pp have said the first thing for you to do is to record what you actually do for several weeks. Yes it’s a pain but you need evidence. Make sure you account for everything, every phone call, email, discussion no matter how small. In particular how much do you do on your 2 non-working days and evenings etc. This will be key evidence to support your business case to retain part time working. Additionally, as a pp has suggested if your Director wants you to take on extra work for increasing hours you can prove you are already at/over capacity.
At the same time work out your non-negotiables e.g. 2 nights with no evening working, one non-working day, flexibility to pick up children, etc.
Once you know the true hours you are working and when, you can then start to work out a flexible working pattern. If necessary come back on this thread and we can help.
As pp have said also look at the team structure and identify one or two posts/team members who could be deputies. This makes sense from a business continuity perspective e.g. illness, holidays etc. It also makes sense from a motivation perspective e.g. learning and development to get them ready for their next step. You could also suggest the deputies get slightly extra pay by taking the difference between your full time and part time salary - I would do this by creating a daily allowance they get on days they cover your NWD.
You could also suggest a 3 month trial period of increasing your hours and days. E.g. increase to 30 hours a week, guaranteed finish at 14:30 on 4th day and deputies cover on day 5.
Ultimately an employer has the nuclear option of consultation, dismissal and re-hire. Not good employment practice, but not many people can afford to go without wages for 2+ years waiting for a constructive dismissal ET.