please empty your brain below

Soon (well, in a few years) to be 0m for me - HS2 will run directly under my living room.
I'm afraid I don't think number 4 is a canal tunnel, only a long bridge.

If it was "officially" a tunnel it would have notices about how long it takes to pass, turning gas appliances off etc.

On your actual question, I happen to have a small disused railway tunnel 2.2km away, but otherwise it would be a long way to go.
I very much fear that the Bow Flyover is a bridge. The waterways maps show it as such, and doesn't a tunnel (even cut and cover) imply some element of excavation in its construction?
Nick beat me to it while I was consulting CanalPlan!
Our flat is directly atop the Northern Line Charing Cross branch and we can feel the rumble of the trains even at third floor.... not sure whether we should be worried‽
My two closest tunnels are both on the Central line. In first place is the full-size tunnel between Grange Hill and Chigwell stations; runner-up is the tunnel immediately south of Newbury Park station where the line disappears underground. In third place is the A12 cut and cover tunnel beneath George Green at Wanstead.
The Canal and River Trust's official website, on its page about Bow Free Wharf, includes a photograph with the associated text "Photo of Bow Roundabout Footpath tunnel".

I recognise that this is extremely flimsy evidence.
The little known Feltham Calvary Tunnel by Hounslow Heath is about 3 miles from me. As a schoolchild is was fun to explore. Other than that Piccadilly Line to Heathrow tunneled section is also nearby.
The Crossrail tunnels were actually given special approval to run at a very slightly steeper gradient than is usually allowed on a railway line in order to be able to pass above the Ham and Wick sewer and thereby remove the need for a very costly and very disruptive diversion of the sewer pipes, this approval also removed the need for substantial lane closures that would have had to be put in place on the A12 passing above the works, in addition to the gradient change works to strengthen the sewer at the point where the tunnels cross over it also took place.
We haven't got mains drainage here, so not much chance of a tunnel.
Assuming there are no large sewers around, and that the District Line’s brief trip under the A316 doesn't count, I think my nearest tunnel is the former Palm House Railway in Kew Gardens, a little under two miles away.
Tube tunnel between East Putney and Southfields. A great viewing point as there's a bridge over the portal entrance with views south along the track.
I've got several tunnels within easy walking distance. The nearest is probably part of the wartime air raid precaution tunnel network under Ramsgate, the end furthest from part which is normally open to the public. But there is also a nearby road tunnel, a little known smuggler's tunnel, disused rail tunnels and chalk mines. And a newish box-push road tunnel passing directly beneath a rail-road level crossing.
First thought was maybe 10 miles or so to a tunnel on the Settle-Carlisle line.

But Upper Swaledale was the scene of some of the most intense lead mining for several hundred years up until the early 1900s. So I suspect a man-made tunnel, probably now inaccessible, is within a mile of here emanating from the Beldi Hill lead mine (subject of mine wars and a court case in the London courts).
Nearest (150m) the Bull Brook which dives into a culvert made many years ago. But trying to walk though that not recommended. Otherwise just a few pedestrian subways which wouldn't count (300m nearest). No real tunnels for many miles.
Chelsfield tunnel, near the summit of the South Eastern Mainline.
As a traditionalist, a tunnel has to be dug/bored through the surrounding material, so the Bow Interchange is mealy a covered trench (cut and cover like much of the District Line), the Abbott Road 'tunnel', Lea Navigation and DLR/Crossways Estate are wide/long bridges, and the Northern Outfall Sewer is more of a pipe.

But as shown by your visit to Abbott Road, there comes a point where it becomes a 'tunnel' for reasons of lighting, fire regulations and ventilation.
Ah, so a cut-and-cover "tunnel" is not a tunnel...? That makes an easy answer (I live reasonably close to the cut-and-cover rail structure in Ryde, Isle of Wight, but would be hard put to name the next-nearest example of a "tunnel".
I wonder which point in Londonn has the most tunnels a mile or closer? 13 is a good start!
I suggest anyone drawn to tunnels takes a look at Auckland, New Zealand. There, not only did a huge motorway tunnel project, the Waterview, take place recently, but the railway system is undergoing a major addition in the city centre at the moment. Not bad in a country known for earthquakes and volcanic eruptions!
The nearest tunnel I know of is the sewer in Kingston where a giant fatberg was discovered in 2013, about half a mile away. Of tunnels to which the public have access, there are several about half a mile away, one being a department store built over a dual carriageway, but the nearest "true" tunnels, dug through the ground rather than bridged over, are all about 5 miles away, at Hounslow West, East Putney and Morden.
If CRT call it (no.4) a tunnel on their website, then fair enough for including it. Honestly though, does it look like a tunnel?

Meanwhile my nearest, I guess, is a much shorter underpass beneath the A61.
I'm virtually on top of the Northern Line (Edgware branch), if I was to bore down it would be 10m to one side.
How would we know if there are sewers nearby?
I know I have a main(ish) sewer in my back garden as it came back as part of the searches when we bought the property.

I suspect it is person sized and probably goes into the "earl sluice" peck branch. Other than that until the bakerloo line extension turns up I suspect my nearest tunnel will be the overground as it ducks under at Surrey Quays.
If you do Urbex in North London, does that make you an Urbexer (NL)?
My dad's old place in Stockwell was directly above the line between Brixton and Stockwell stations. I'd guess it was akin to a very minor earth tremor every 2 minutes or so, no noise at all, but a reassuring deep bass rumble that only really became noticeable by its absence after the last train had passed through.
I was going to argue about what is and isn't a tunnel, but I see that's been covered already!

So instead, probably more to DG's liking, I'll just answer the question: my nearest tunnel is the Dreadnought drain aka Malago Stormwater Intercept. Urbex types have explored it, I have not, being inclined to say more on the "Mild Urbex" side of the fence.
Rotherhithe, Brunel's tunnel and the Thames Water super sewer are the main ones, and if DG is counting that bridge, then I'm counting the one at the Wapping ornamental canal. Crossrail is a little further away.
Greenwich foot tunnel just around the corner.
It was an interesting experience to walk through the day (few years ago) the power failed and the emergency lighting also failed.... A few souls picking our way through using mobile phone screens for lighting.
What's impressive about this post is the level of research that DG has gone to. I realise that at home he's probably surrounded by maps and plans of every description, but knowing where to look to discover the route of a sewer, rail line or anything else is impressive. Well done Sir!
Is the pylon replacement tunnel a contender?

dg writes: Added, thanks.
This is my nearest tunnel.
Woolwich Arsenal DLR is my nearest tunnel. Also the Woolwich foot tunnel. Then I think the next one is Blackwall tunnel and North Greenwich tube.
Excellent tunnel research, notwithstanding the "probably a bridge" comments.

I live in Brixton and I would love to think that some part of the "lost" River Effra (with which I have recently been fascinated and/or obsessed) still flows in a tunnel (or sewer) near me. There are a few stinkpipes nearby but I suspect they are they relics from stinky times past and that the river has long since been diverted into sewers upstream.

A couple of years ago I did find your blog post on the pavement plaques commemorating the River Effra and have followed its course and found 35 in total myself. I did email Lambeth Council to ask if anyone knew how many there were in total but nobody seemed to know.

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