The first live radio recital by a professional musician took place at the Chelmsford factory, by the legendary Australian diva, Dame Nellie Melba.
Earlier experimental transmissions by Lee de Forest (an "October day" in 1907), and reportedly an even earlier one by Reginald Fessenden (1906), had shown the technical feasibility of broadcasting live music for entertainment. However the quality was not deemed good enough for a commercial service - see Initial broadcasting experiments.
# When was the first?
There is no clear date we can put our finger on with regard to the first ever live public music performance on radio. There were a number of experimental broadcasts around the world, many of which featured live performances.
Lee de Forest is a solid candidate fo the first public radio broadcast of a live singer.
January 13, 1910 Lee de Forest - live opera broadcast from New York.
On January 13, 1910, he succesfully demonstrated an experimental transmission of a live performance by several famous opera singers at the Metropolitan Opera House (39th St). Tosca was performed on January 12, 1910, and the next day's test included Italian tenor Enrico Caruso.
Lee de Forest later renewed broadcasting activities on October 26, 1916, as part of an arrangement with the Columbia Graphophone Company.
Beginning November 1, Radio 2XG (the "Highbridge Station") offered a nightly schedule featuring the Columbia recordings, which as far as I can tell can be considered the first regular commercial radio sation to play music.
# Live from Australia
The first broadcast of music in Australia was made during a demonstration on 13 August 1919 by Ernest Fisk of Amalgamated Wireless (Australasia).
13 August 1919 First demonstration music broadcast in Australia
# Live from Chelmsford
On 15 June 1920, Dame Nellie Melba was heard in a pioneering radio broadcast from Guglielmo Marconi's New Street Works factory in Chelmsford, singing two arias and her famous trill.
14 February 1922 First regular wireless entertainment broadcast
If you disregard the Initial broadcasting experiments which were of poorer quality, and not deemed good enough for commercial purposes - she was the first artist of international renown to participate in direct radio broadcasts.
She opened her recital at 19:10 on 15 June 1920 by singing Home Sweet Home and after other popular favourites and several encores, closed with the National Anthem - bbc 
Radio enthusiasts across the country heard her, and the broadcast was reportedly heard from as far away as New York. People listening on the radio barely heard a few scratches of the trill and two arias she sang.
# Live from Argentina
August 27, 1920 Wagner broadcast live on radio from Buenos Aires.
On August 27, 1920, in Argentina, they were finally ready. The theatre was showing the opera Parsifal by Richard Wagner. Enrique Telémaco Susini and his coworkers had set up a 5W transmitter on the roof using a RS-5 Telefunken tube, operating in 350m, along with a wire antenna connected to the dome on top of a neighbouring building -
wikipedia
# Regular entertainment broadcasts
Just over 18 months after Nellie Melba's succesful trial broadcast, 2MT began regular transmissions from an ex-Army hut next to the Marconi laboratories at Writtle.
15 June 1920 Nellie Melba sings live from Chelmsford
Regular broadcasts started on 14 February 1922. Initially the station only had 200 watts and transmitted on 700m (428 kHz) on Tuesdays from 2000 to 2030 - wikipedia
# Empire Broadcast
Further radio broadcasts would include her Covent Garden farewell performance, and a 1927 "Empire Broadcast" (broadcast throughout the British Empire, by radio stations AWA and 2FC, Sydney, on Monday 5 September 1927; it was relayed by the BBC London on Sunday 4 September) - wikipedia