As mentioned in another answer, the uncredited actor is a boy named Michael Jones. This was confirmed in Doctor Who Magazine #519 in a feature about William Hartnell.
The Tenth Planet in October 1966. Since Hartnell's death in April
1975, several actors have given us their First Doctor. Richard
Hurndall in The Five Doctors (1983) was the first to step into
Hartnell's shoes, metaphorically speaking; then came former companions
William Russell and Peter Purves, in various Big Finish audios;
voiceover artist John Guilor, albeit briefly, in The Day of the Doctor
(2013); and Michael Jones played him as a child (probably) in Listen
The Doctor Who Daily Blog on AminoApps (usually a reliable source of information about the show) indicates that he was chosen for his resemblance to a young Hartnell.
Archive footage inserted in post-production indicated that this was
the same barn occupied by the War Doctor in The Day of the Doctor.
Michael Jones was cast as the young Doctor, after the crew noticed his
resemblance to a photo of William Hartnell as a child.
No indication is given as to where they located him from and/or whether he was a relative of the crew or production team (as is common for uncredited child cameos on BBC programmes) or from a talent agency.