please empty your brain below

When I lived in the UK, our house was 900m from Ruislip Woods. As a child on the weekends and in the vacations, my friends and I would play for hours unsupervised.
Osterley Park is well looked after, I think parts of Hounslow Heath by the River Crane are more natural or wilder looking woodland.
Barnsbury Wood in Islington probably doesn't count, but just in case you are not aware of it.
While Richmond Park has two main woods - Sidmouth (Rhododendron) and Isabella Plantation (ditto) - it's use as a deer park means that nothing grows large unless it is well protected.
For a more natural feel I'd pick Ham Common Woods, just over the Park's west wall.
For Hounslow, I'd choose the free Chiswick House and Gardens over the pay-to-enter Osterley.
Poor Newham. Although the northern bit of Wanstead Flats is woodland if they don't want to walk too far.
While Selsdon Wood probably does win for Croydon, King’s Wood in nearby Sanderstead is a decent second with some stunning bluebells in the spring.
Both Sidmouth Wood and the Isabella Plantation are fenced off from the deer, but I would agree Ham Common has the more natural feel to it.
Croydonians are spoilt for choice, and I'd also nominate Croham Hurst and Addington Hills.
I'd offer the following areas as potential alternatives:
Barnet - Hadley Woods
Enfield - Trent Park woods
Update 1: Islington, Hounslow, Richmond
(thanks)
Lambeth: Streatham Common
That’s the area at the top of the hill, next to The Rookery
Small area of woodland 200m away (still meeting your 2 hectares rule). Huge area of Crown forest 500m away. Berkshire though has loads of woodland.
Perivale Wood is a private wood so I would go for Horsenden Hill as Ealing's "best wood".

Osterley Park is free to enter for walkers (the car park isn't free).
Perivale Wood is gorgeous, especially at bluebell time, but as it's only open to the general public one day a year (to coincide with said bluebells), I think it might fall foul of one of your rules. However, just next door, the slopes of Horsenden Hill are pretty wooded - increasingly so in places - so maybe that should be Ealing's entry.
Not geographically in the City of London, but there are plenty of woodland areas managed by the Corporation e.g Burnham Beeches, Ashtead Common; as well as Epping Forest.
Update 2: Barnet, Ealing, Enfield, Lambeth
(thanks)
Newham: seriously clutching at straws here but what about the Memorial Garden within the Bow industrial estate that Cody Dock is located on? This site describe it as a "small section of woodland" but no idea if it's as big as 2 hectares!

Otherwise there's the wooded-ish walk around Abbey Mills but I'm sure that doesn't count either!
In addition to Streatham Common, Lambeth has a small area of woodland in the area of Tooting Bec Common within its boundary, near St Leonard's church, Streatham.
I'm not convinced I'd put Highgate Wood at no.1 in Haringey. OK, it is well accessible and Corporation of London maintained but Queens Wood I think is more amazing, closely followed by Coldfall Wood.
Queen's Wood Islington
Nice picture.

2 ha is pretty small. Hard to get much fun walking round a 140m by 140 square. A strip 35m by 560m might be slightly better I suppose.
Failing all else, for City of London I'm going to nominate the Barbican Wildlife Garden on Fann Street. Whilst it's small and most of it is open there's a lovely strip of woodland with a path running through it that separates it from some of the neighbouring buildings. Maybe I'm clutching at straws, but I visited during Open Garden Squares and you can (for a moment) truly feel you're in a proper woodland and not in the heart of a large city.
While it's nice to have woods on your doorstep (as I do), in London it's very easy to get out to visit woods elsewhere in the city. Indeed that's why it was such a good move in Victorian times that the wealthy City of London corporation would manage open space elsewhere in London for the benefit of everyone.

For Westminster, I think parts of Hyde Park might just about count. The area north of The Serpentine and to the east of the Sackler Gallery does have a fairly decent tree cover and is surprisingly quiet

I'd agree with Holland Park for K&C
I am between 7.5km (Ruislip Woods) which I think of as being my nearest woods and 9km (Stanmore Common) which is in my borough, so I'm astounded to now discover that Horsenden Hill is actually the closest at a mere 5km, as that's the one I always ruled out as being to far away!

Either way, all involve at least 2 buses or a bus/tube combination, so none were of any use during the lockdown.
Hainault Forest straddles the borders of Redbridge, Havering and Essex, and it is set to grow larger (although not as large as it was before disafforestation in the 1850s). As part of the Mayor of London's bid to improve access to the Green Belt, the Woodland Trust has recently been awarded nearly £500,000 to plant a further 35,000 trees across 26 hectares there.
Not a wood, but Finsbury Circus is probably the closest thing in City of London! All the other areas of dense tree'ing are either private (Barbican) or rows (like the half of Temple garden in the City of London boundary).

Hounslow is absolutely that stretch of the Colnbrook river, near the Heathrow balancing ponds and round that way are all pleasant.

I also found it amusing that 'Waltham Forest's wood was Epping.
Hammersmith & Fulham - Wormwood Scrubs has some woodland
No mention of Wanstead Park? When I was growing up in Forest Gate we thought of that as the jungle
Update 3: Hammersmith and Fulham, Haringey
(thanks)

A lot of you have suggested woods I nearly picked, so I've been happy to switch.
I have always thought of Putney Heath as essentially part and parcel of Wimbledon Common, with only a busy road to separate them.
Ealing's website has several 'woodland walks' but, other than Horsendon Hill, they're mostly tree-lined strips and patches. Gunnersbury Triangle near us fits the bill at 2.5 hectares of neglected then conserved woodland trapped between railway lines.

I too would gve Hammersmith the wooded bits on Wormwood Scrubs.
Not sure of the criteria for the “?” but the woodland bit of Streatham Common is marked as such on the Ordnance Survey and, by my rough measurement, is around 4 hectares. Together with The Rookery and Norwood Grove it’s worth a trip.
Harrow:
I recommend Stanmore park / Bentley Priory open space which includes "The Master": the oldest tree in Middlesex.

Stanmore country park is a separate park but is also excellent woodland area
Oaks Park in Sutton. I think the wooded areas meet the 2 hectare rule.
Living within 200m of two different woods (Spa Hill Woods and Grangewood Park, both parts of the older Norwood) I must be spoiled, although they are both small. But yes, Croydon.

Yeah, I think the woods at the top of Streatham Common qualify, although they're not very big.
Queen Mary's Woodland in Sutton is good.
For Newham; Would the 'East Ham Nature Reserve' count?

dg writes: Answer in post.
Any further thoughts on Harrow, Kingston and Sutton particularly welcomed.
Ah, wasn't sure if that was the same or not. The 'Alps' below it appear to be closed off looking at google reviews, but that is about same size and more natural.
Kingston - not much to choose from though much of Sixty Acre Wood isn't in the Borough I think. An alternative is Coombe Hill Wood though it's only 5.5 acres and its proximity to the A3 doesn't help its attraction. On the opposite side of the A3, you have a wooded part of Wimbledon Common which is in Kingston which I'd choose but only because it joins onto larger Fishponds wood in Wimbledon Common (which is in Merton).
Kingston University has a very pleasant, but small, forest on its Kingston Hill campus, and indeed runs a 'forest school' there for children with social anxiety. Am not sure what the rules are for public access, though.
Update 4: Kingston

Sixty Acre Wood turns out to be private, so Coombe Wood (divided by the A3) looks to be the borough's best, thanks. But poor show, Kingston, poor show.
I should make up for the lack of Bromley representation in the comments by saying how much I used to enjoy exploring the woods on Hayes Common as a kid.
Selsdon Wood is - in my view - beaten by Addington Hills, which is larger and more interesting. Selsdon Wood is also a little lost behind nondescript housing estates whereas Addington Hills has Lloyd Park and Coombe Wood nearby.










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