The June issue of Underground News includes an article under the above
title by W.R. Lynch, which begins as folllows:

# When a new record was announced in 2005 of 18h 35min 43s to visit
# every LUL station, it sparked me off to analyse the logistics,
# survey the critical points and have a go myself. I failed by
# 35min 55s to beat the record, and, although it was an interesting
# project and I got some satisfaction from completing the course,
# I had always felt that the all-stations bash was a less than
# satisfying exercise in that it left odd parts of the network
# not traversed, and I couldn't raise the enthusiasm to try again.
#
# What did nag away at my thoughts, though, was the observation
# that the record was achieved in 2h 16m 17s less than the traffic
# day of 20h 52min. Could the skipped network sections be fitted
# within that 2¼h to make an all-lines bash?

Lynch decided that the correct rule was for lines sharing tracks or
running side by side to be counted once, so only one trip would be
required between Barons Court and Ealing Common, for example; and
that non-Underground trains could be used on such sections (e.g.
Silverlink from Richmond to Gunnersbury), although this turned out
not to be needed. He or she also decided that it was reasonable to
exclude the Watford north curve, which trains use only at the start
and end of the day and which does not appear on the system diagram.
The Heathrow loop was, of course, unavailable.

Now, the Guinness Book people have a set of rules for visiting all
stations, which include the requirement that to get from one station
to another one must either walk or use scheduled public transport.
Lynch could not construct an itinerary using that rule, so he or she
decided to additionally be allowed the use of a bicycle. And since
folding bikes can be carried without restriction, that was the trick.

Lynch's first three attempts, one each month from December 2005 to
February 2006, were failures, but informative ones. The fourth
attempt was on March 7, 2006, and it succeeded. Train 214 left
Hammersmith at 04:39:35 with Lynch and his or her folding bicycle
on board. And 51 trains later, Lynch arrived at Upminster at
01:20:08 on train 104 -- not the last, but the second-last train
of the day. That's 20 hours, 40 minutes, and 33 seconds. Lynch
reckons the distance as 509.7 km, or 2.67 pence per km on a £13.60
Travelcard...

Rather than carry the folding bike for the whole trip, he or she left
it parked at 16:30 at Stanmore. There were two more bicycle legs
after that, but an ordinary bike had been left at Harrow & Wealdstone
in advance; it was used for these, between about 21:30 and 22:10,
and parked in turn at West Acton. (Lynch says that switching bike
proves not to be worth the trouble, though.)

Lynch concludes:

# I recruited no witnesses and took no photographs. So what?
# If you prefer not to believe me, that's find by me. I'm also
# quite happy for you to attempt the same challenge to whatever
# standard of proof you like, or to adjust it to suit yourself --
# my rules are not law. Good luck!

Lynch's route, if I haven't miscopied it, was:

Hammersmith, Edgware Rd.;

Paddington, Gloucester Rd., High St. Ken., Wimbledon;

Morden, Tooting Bdwy., (via Charing Cross to) Mill Hill E.,
Finchley C., High Barnet;

Cockfosters, Barons Ct.;

Ken. Olympia, High St. Ken., Edgware Rd., (clockwise to)
Tower Hill, Aldgate E., Liverpool St., White City, W. Ruislip;

Ickenham, Uxbridge, Harrow-o.t.H., Amersham;

Chesham, Chalfont & L., Moor Park, Watford, Moor Park, Baker St.,
King's X. St. P., Walthamstow C., Brixton, Stockwell, Kennington,
(via Bank to) Edgware;

Stanmore, Wembley Park, Finchley Rd., Canada Water, Shoreditch,
New X. Gate;

New X., Canada Water, Stratford, Epping, Woodford,
(via Hainault to) Bank, Waterloo, Elephant & C., Harrow & W.;

Rayners Lane, Ealing Common;

W. Acton, N. Acton, Ealing Bdwy., Acton Town, Heathrow T.1,2,3,
Turnham Green, Richmond, Upminster.

Each "paragraph break" indicates a foot or bicycle segment.
--
Mark Brader | "The race is not always to the swift,
Toronto | nor the battle to the strong --
| but that is the way to bet it." --Damon Runyon

My text in this article is in the public domain.