The singer was Millicent Small, and she recorded the song under her stage name of Millie.
The song was a massive hit in March 1964, reaching number 2 in the charts. Although she had a couple of minor hits, Millie is essentially a "one-hit wonder" and nothing has been heard of her lately. Her one big hit was produced by legendary producer Coxone Dodd, and he was assisted by Chris Blackwell, founder if Island Records who discovered Millie and brought her to the UK. Blackwell's other major discovery was the equally legendary Bob Marley.
If the rumours are true, the harmonica on 'My Boy Lollipop' was played by one Rod Stewart!