I live in London zone 2 and I dont drive, but I would like to learn when I have more time. Personally I find that it's not at all essential, but there are certain things I could do with a car which I can't do/are harder with public transport:
Certain holidays, eg Cornwall and other parts of the UK are hard to do without a car. We've done it but not been able to do certain excursions.
Big shops, especially to places like Ikea as their delivery costs are high (we just pay for delivery)
Some day trips eg NT properties and even places in outer London are hard to get to
Extracurriculars for dcs - would be quicker with a car and easier for them to rest in a car than a bus then tube
Picking up secondhand items eg from Ebay - I just don't buy certain things that are collection only, although I've managed to buy some large items and bring them home on the tube.
So mostly it would help with leisure activities or discretionary purchases, and so far I've managed without driving for important things (work is a 20 min walk and school is 5 mins). But I plan to learn once my youngest is in school so I'll have some free time.
I did try to learn to drive a couple of times in the past, and each time I spent well over £1.5k, because I'm dyspraxic and find it hard (and that was more than a decade ago so prices will be higher now), and I ended up failing both times. So I'd expect it to cost far more than that for me to learn to drive, although you may be much quicker. Like you I didn't have anyone to help me practice, so all my practice time was through paying for lessons, and it does add up. Insurance will be very expensive too for a new driver. So I'd make sure you can set aside a decent chunk to pay for lessons and all the other costs, because once you've started you have to keep going or you'll forget it all.