We have four children but a similar sort of situation otherwise. Due to needing four lots of after school club (or even four lots of childminder) plus DS has SEN and often challenging behaviour, we opted for a nanny. Not just better value for money, but also better for the children to just go home and chill (or be taken to their clubs etc).
Not going to say it was all sunshine and rainbows though. While yes, it is very possible to get after school nannies as we did, as has been mentioned you're more likely to get ones with their own children. I suspect this is partly because they often find it harder to get work so are more willing to be flexible with hours. However, if their children are school age, then there can be logistical issues with collecting theirs and your children unless they can put their own children in clubs/with family etc. And you have to consider what happens in the school holidays and find something mutually agreeable.
Also, one thing I didn't initially appreciate was the overhead of time involved in the bureaucracy. Not just interviewing and contracts, but calculating holiday pay (especially as hours varied, particularly in school holidays), and the three nannies we had all got pregnant at some point too. So then there's SMP.. and although the government pays that, it doesn't pay their holiday pay which they are legally entitled to, so that's another 28 days pay (pro-rated if necessary) if they take a full year out and meanwhile you're paying for cover in other ways. And pensions! You have to set up a pension holding account regardless, giving them the option of joining, and once they earn over £833 a month, you have to sign them up to it unless they say not to. There's the quarterly tax bills to HMRC you have to pay. You need to make sure you have employer's insurance. Then there's sickness - what is fair in terms of sick pay? What if their child is ill and they can't work? What if they give you nearly zero notice that they can't work that day (as happened with us on many, many occasions?)
Although the payroll can be covered by a nanny payroll company with an annual fee, there's still lots you must handle yourself. I became an HR expert pretty much.. had to be.
And although nannies do become very attached to the children they look after, and even if you treat them really well (I know I was a soft touch and one in particular really took advantage) in some cases, like with any career, they may well be looking or be approached with better or more suitable opportunities. Then you have to decide whether to match it or not. I had a 4% payrise at work over three years, my nanny had around 25%.. but I couldn't stand the hassle of finding a new one each time, and didn't want to upset the children with change.
There are huge positives though. The children were a lot happier than at after school club. They still did their external clubs e.g. swimming, gymnastics. Homework got done. We only requested nanny housekeepers, so the children's washing got done and put away, and toys got tidied away. They had tea made. They went for fun walks to the park, picnics, local farms etc. They had somebody willing to listen to monologues about Minecraft. They had great friends in the nanny's other child/ren (yes, this is often a big plus of a nanny with her own child). And for the most part, aside from pregnancies and house moves (the only reasons we lost nannies in the end) there was a level of stability you don't always get with other childcare settings.
As of last week, we finally no longer have a nanny. The final one is now on maternity, and was down to one afternoon a week anyway as I recently took a job working from home. It may well be as a result of my job change that we no longer need her post maternity leave. I will be massively relieved to lose the headache of all the admin/HR stuff which you have to do on top of full time work, commuting, and being a parent. Most of all, I'm relieved to lose the anxiety over hoping the nanny doesn't call in sick/has an emergency, or worse, wondering if the nanny will stay; you're at the mercy of somebody else's career or location decisions.
However, it was the right solution for our children despite the extra burden and stress it caused me. And our nannies (three in total, sometimes we had two at a time job-sharing) were all lovely people, and I still have a nice friendship with them all.
We got all our nannies through childcare.co.uk - make sure you check references etc but it's a very good source.