Alan's note: All about attitude.  Thumbs in braces or hands on lapels,

Lambeth Walk Sheet Music Instructions, described by Arthur Murray:

1) Partners march side by side, gentlemen on the left. Strut forward 8
steps (4 bars); swing the arms, walking jauntily in cockney fashion.

2) Link right arms, walk around in circle to right 4 steps. Quickly
reverse, linking left arms, and walking 4 steps in circle to left.

3) Strut side by side again 8 steps (same as figure 1.) Partners
separate, face each other, taking 4 very short steps backward. Close
heels on 4th count.

4) Slap knees in time to music.

5) Ending with pointing thumb over shoulder, in hitchhike fashion; and
yell loudly, "Hoy!" Repeat from beginning. It is necessary that the
steps fit the music. The dancers should start on the very first beat
of the chorus. 

Alex Moore's book:

Couples start side by side, about arms length apart; man Left Foot, lady Right Foot.

8- walks forward, turn to face partner at the end of walks.

8- walks in a circle, L arms linked. At the end, unlink arms, turn
side by side and link mans R and ladies L arms.

3- walks, rock backward and forward. (Count 1, 2, 3, & 4, similar to
an understated samba walk??), repeat 3 walks and rocks on other foot.

2- walks away from partner, turn and slap knees with both hands (count 1,2,3,4)

2- walks towards partner, turn to original starting position and a
single jerk of the thumb, shouting "Oi". 

How It's Done (1938 Sun Newspaper):

Everyone gets in a big circle with each couple side by side, and arm
in arm. To the tune of the Lambeth Walk they sort of strut around the
floor. At one part of the song each does a semi-circle and the arms
which have hitherto been free are the ones that are linked. This
movement takes place three or four times in rapid succession. It is
the difficult part of the whole dance. Doing it swiftly and gracefully
(?) and yet keeping time with the music is no small assignment,
although the Big Apple addicts should find it simple, maybe even too
tame.

If the song is played about as long as is the average number, there
are about four times during the dance when the couple are bent over
and forward, and directly facing each other. To the rhythm of the
music they then clap their hand on their respective knees, and then
all simultaneously throw their hands in the air and yell "Oi!" Then
the strutting around the floor in circle form starts all over again.