The noun patter denotes the sound of light footfall, and the phrase the patter of tiny feet, and its variants, denote the presence of one or several young children, or the imminent birth of a child.

Prior to the earliest occurrence of the phrase, “a Correspondent” used the metaphor in the following announcement, published in the Aberdeen Journal, General Advertiser for the North of Scotland (Aberdeen, Aberdeenshire, Scotland) of Wednesday 4^th August 1819:

Married, at St. Andrew’s, on the 21^st current, Mr. David Niven, flesher, to Miss Margaret, eldest daughter of David Wright, Esq. Writer there. St. Andrew’s has long been proverbial for the number and celebrity of maiden ladies belonging to it, and the rare instances of any matrimonial union occurring within its precincts, but the times are now altered; for, within the last 12 [?] months, this is the fourth marriage that has been celebrated in the same row of buildings (consisting only of five dwelling-houses, and to be henceforth styled Union Place); and if report speaks correctly, more are likely soon to follow. Were the same honourable spirit for matrimony to pervade the other districts of the good old city, another venerable feature might soon be obliterated—the long grass on the streets, which for many years has been kept down only by the scythe and the “town’s nout,” might be destroyed by the pattering feet of pretty girls and chubby boys.—From a Correspondent.

The earliest occurrences of the phrase the patter of tiny feet and variants that I have found are as follows, in chronological order:

1-: From York Minster; and a Chronicle of the Times of Wickliffe, published in The Church Magazine (London, England) of August 1842—reprinted from The New York Churchman:

It was not many years before the death of Edward the Third, that there resided in the city of York, a famous master of handicraft—one Walter Gower by name, and a sculptor in wood and stone by profession. A cheery, pleasant man was Walter Gower. His artistical skill no man could gainsay; whilst all his neighbours proclaimed him a frank-hearted soul, and a good citizen.
Particularly cheerful and merry was Walter on the present occasion. It was a bright, fresh morning in December, a day or two before the festival of Christmas, and he had walked out into a neighbouring forest, to gather some branches of ivy and holly to decorate his dwelling. He had accomplished his task, and was returning homeward, laden with an armful of glossy ivy and green holly, humming, as he walked briskly along, the burden of an old Christmas carol. In the midst of this glee and good humour, his ear was suddenly awakened by the sound of something approaching his path. It was the quick patter of tiny feet, together with a childish voice, beseeching him to stay his footsteps “for mercy’s sake!” He looked around. Walter was a humane man, and a cry of distress was enough to excite his sympathy at all times. He instantly halted until the little messenger of grief—almost breathless—came up with him. In another moment the child was at his side. And seldom had he gazed upon a more lovely or engaging countenance than that which now looked up imploringly, whilst she besought him to come to her poor mother, who was dying hard by.

2-: From Mrs. Newington, an unsigned short story published in The Northern Whig (Belfast, County Antrim, Ireland) of Tuesday 18^th April 1843:

Mrs. Newington having informed her niece, that “servants were the greatest pests in the world, and that nobody ever had such a plaguy set as herself,” set out upon her journey to her sanctum sanctorum, but stopped at the drawing-room landing place, on hearing the patter of little feet, and the sound of childish voices within. Now, Mrs. Newington’s drawing-room was a paradise which her little angels were never permitted to enter, except when handed in for company commendation, and handed out, after the flattering dose had been sufficiently administered; she therefore opened the door with considerable alacrity and apprehension.

3-: From The Young Bride’s Trials, published in The Gossips of Rivertown; With Sketches in Prose and Verse (Philadelphia: Hazard and Mitchell, 1850), by the U.S. author Alice Bradley Neal (1828-1863):

Home-sickness—it was the first real pang she had found leisure to feel since her marriage—was added to her unhappiness. This was her home now, it is true, but how unlike the cosy little parlour at the cottage; and her mother’s gentle smile would come side by side, and in sad contrast to Miss Margaret’s immovable face, as often as she looked up. Where, too, was the patter of little feet, the sweet murmur of children’s voices? She wondered what Willie, and Etta, and Harry were doing now!

4-: From A Western Tale. Little Fred, the Canal Boy, by the U.S. author Harriet Beecher Stowe (1811-1896), published in the Cincinnati Liberty Hall and Weekly Gazette (Cincinnati, Ohio, USA) of Thursday 14^th February 1850:

He heard gay laughs, as groups of merry children passed, and then he started as he saw some woman in a black bonnet, and thought she looked like his mother;—but all passed, and nobody looked at him—nobody wanted him—nobody noticed him.
Just then a patter of little feet was heard behind him on the flag stones, and a soft baby voice said, “how do’oo do?”—Fred turned in amazement, and there stood a plump, rosy little creature of about two years, with dimpled cheek, ruby lips, and long fair hair curling about her sweet face. She was dressed in a blue pelisse, trimmed with swan’s down, and her complexion was so exquisitely fair, her eyes so clear and sweet, that Fred felt almost as if it were an angel.