Thanks to Andy Kelly we now have information about Arsenal Reserves in the early years.
A new article has appeared on Wikipedia, and this has allowed me to cross reference with other sources, and we seem to have a more clear idea of what happened to the reserves in the early years.
Wiki says that Arsenal Reserves were set up in 1887, initially playing friendlies and cup competitions. In 1895 they joined the Kent League, but were too strong for that league, and left around 1900.
They then went into the London League and won three championships and also won three championships in the West Kent League, although the dates seem to be a bit confusing – but that was all around the end of the 19th centuiry and the early part of the 20th century.
What is clear is that playing the reserves in two different leagues at the same time was not unknown and certainly not frowned upon, which is why sometimes dates overlap.
In 1903 Woolwich Arsenal went into the South Eastern League and stayed there to football closed down in 1915 because of the First World War. It seems that during this period they also played in the London League First Division – although not in every season. Wiki states seasons 1906/7, 1907/8, 1908/9, 1913/14 and 1914/15.
So the consistent point was the South Eastern League – and indeed reference to other clubs shows that they too were putting reserve teams into this league – indeed Tottenham Hotspur still feature the fact that they won it a few times in this period on their web site.
Thanks for this information – slowly the picture is becoming clearer.