This is what PJS Richardson says about the one-step in his "History of English Ballroom Dancing":
"About 1910 the two-step which had come to us from the States, and in which the chasse in all directions was used, began to give way to the one-step . . . when first introduced [it] was a simple walking step in all directions, and James M. Gallatly's "Bogey Walk" was the first tune deliberately composed in this country for the dance."
"The first 'Informal Conference' of Teachers of Ballroom Dancing ... was held in 1920 ... the meeting then considered the fundamental steps suggested by the committee:
One-step: Pas marche, chasse
Foxtrot: Foxtrot walk, chasse. three-step
Tango: Paseo, Corte, Marche Argentine, Carre, Huit"
[Note that they didn't seem to find it necessary to standardise the waltz.]
"In December 1922 ... there were held, for the first time in this country, the World's Championships .... the championship dances were the waltz, foxtrot, tango, and one-step ... [the winners were] Mr Victor Silvester and Miss Phyllis Clarke."
"By 1925 dancers were tiring of the monotony of the one-step and were turning their attention to the quick-time foxtrot, so although in 1925 the one-step was included in the championships, a special competition was held in quick-time foxtrot or quickstep ... in 1925 and again in 1926 the newly established Star Championships kept to the old four - waltz, foxtrot, tango and one-step; but in 1927 the one-step was finally dropped and a new dance christened on the programme as 'quick-time foxtrot and charleston' was introduced ... from which it will be gathered that the modern quickstep, from which the charleston gradually disappeared, was rapidly taking shape and that the one-step was extinct. "
John Nott, a retired government minister, described his attempts at ballroom dancing in "Mr Wonderful Takes a Cruise". He is attending an event organised by dance teacher Mr Wonderful in 2002:
"I protest vigorously that I am incompetent. 'No,' she says, 'you must try.' So I am hauled to my feet and dragged protesting round the dance floor. It is a foxtrot. Afer I had twirled her expertly round the floor, I ask her how I am performing. 'You are doing what is known as a one-step,' she says. 'It went out in the 1920's; you badly need lessons.'"
I sold my collection of old 78's over 30 years ago, but I remember that the tunes described as one-steps were about the same speed as a modern quickstep, with a bit of an oompah sound.