I would take the £15k now, put it in a high interest account.
The £4k will probably get used paying for sperm and IUI unless you have separate funds for that? It can be done unofficially with a guy from a website and a turkey Baster but it would IMO be madness to risk that.
So I would start contacting fertility clinics and getting quotes and timescales. Let’s say it will be 3 months to sort everything with the clinic, even if you get pregnant first go you’ve got 9 months plus work to your due date and get 6 months full pay. You could save £5400 plus interest in that time plus get the interest of £1000ish from the £15k if it’s in a 5% account during that time. You will also have child benefit but factor using that for nappies, wipes etc (reusables are cheaper and environmentally friendly but ime much tricker especially if you’re doing it alone). All other baby stuff can be found second hand for free/ at low cost. If in Scotland you’ll get the baby box which helps a lot.
So the key thing is post 6 months of mat leave what are you going to do? I would take advantage of workplace policy on annual leave and if allowed to, try not to take any whilst pregnant and tack on to the end of mat leave, keeping a week or two for inevitable sickness.
Look at childminders who can be much cheaper than nurseries and consider flexible working options that might mean you don’t work 5 days a week but don’t lose much salary. If you have £400p/m from the £15k, the £300 you would otherwise save plus £250p/m from your other accrued savings, plus tax free childcare, you might find a childminder who can do the hours you need for around that. Our DD does 2 days at a council nursery for £330p/m (once tax free childcare accounted for) which would be £825 full time, so depending where you are there may be affordable options.
At the 3 year mark you’ll likely have depleted all savings so you’ll need to use the 30 hours free childcare and find an option where you can get the remaining hours for £300ish.
For school our local childcare place does summer holidays for £35 per day, or £140 per week with tax free childcare, potentially affordable if you alternated with annual leave. The after school care is £52 per week with tax free childcare.
Look into all of this, you’d have to really cut your cloth but if you really want a child it is probably doable.