Following discussion on Ringing Chat and Change Ringers email lists in February 2008, the following is believed to be the list of the oldest ringing societies still functioning as a ringing society.
It has to be acknowledged that over a period of approximately 400 years any organisation will have had its high and low points. The dates shown are for generally accepted original formation.
1612 - The Company of Ringers of the Blessed Virgin Mary of Lincoln
The Company of Ringers of the Blessed Virgin Mary of Lincoln was formalised on 18 October 1612, in a constitution granted by the Dean, Laurence Stanton, and his Chapter and is the earliest known Company of Ringers still performing their duties for which they were set up. 
1623 - Saffron Walden (Essex) Society
Ernest Morris - "This is the oldest local company of the present day". However this date is disputed by the Society itself,  which states that whilst ringing has been carried out for many years the Society as such has probably only existed since around 1800.
1637 - Ancient Society of College Youths
Established in 1637 based in the City of London, with a national and international membership that promotes excellence in ringing around the world. First peal - 1690.
1672 - Society of Sherwood Youths
The origins of the Society are lost in history but there are references to its establishment in 1672 and it may well be that it was the successor Society to the 'Northern Youths' who gave the two new treble bells making the octave to St Peters Church Nottingham in 1671. First peal - 1762. 
1686 - Ancient Society of Painswick Youths
All the band who qualify to join the The Gloucester & Bristol Diocesan Association of Church Bell Ringers are automatically members of the Ancient Society and two dinners a year are held currently funded from wedding money. A stone tablet in the ringing chamber record the first two peals in the tower "A whole peal of Tripples of 5040 changes" on 19th February 1731 and again on 1st March 1731. [Information provided by Angela Newing]
1724 - Society of Cambridge Youths
The SCY are the second oldest ringing society at any church with a continuous history as a ringing society after the Lincoln Cathedral Guild of ringers.
1734 - Oxford Society
Considered to have been founded at the time of the ringing of the first peal in Oxford, Grandsire Caters at Christ Church by a local band.
1747 - The Society of Royal Cumberland Youths
The Society of Cumberland Youths was founded on September 6, 1747 - the word "Royal" being added in the 1870s.
Old Societies No Longer in Existence
1620 - The Antient Society of St Stephen's Ringers
This society does still exist today but has long since stopped functioning as a ringing society. It has a membership limited to exactly 100 members, and continues to support St Stephen's church financially.