Early Lee Enfield
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The "father" of your rifle was the Lee-Metford. That is, the Lee action combined with the Metford type of rifling. When the British changed to Cordite as a propellant, they found out that the higher temperatures of the Cordite burned out the Metford type rifling quickly.
In 1895, they adopted the Enfield type rifling to give a longer barrel life. The Lee Metford rifle was essentially fitted with a barrel of this type, thus becoming the Lee (action) mated to the Enfield (rifling). As stated, the rifles (and converted Lee-Metfords that were rebarreled) have the letter E on the Knox form of the barrel.
Makers of the military rifles also provided private citizens, Regiments, and others with the same pattern rifle as the British Army. These were marked with the makers name (in your case, BSA) but do not have the government ownership marks (eg VR, GR, and broad arrow) or a crown stamp. Some rifles were approved for use on the range as Military standard, after being inspected by the Armourer to make sure the rifle conformed to Military specifiactions, and were then stamped with a second number signifying this approval.
In 1900, a Small Arms Committed was formed to try to improve the weapon from lessons learned during the Boer war. In December, 1902, the Short, Magazine, Lee-Enfield was approved. This was the SMLE Mark I and was modified in 1907 to become the SMLE Mark III
You state that the sights are re-stamped. Many of the earlier Lee Enfields and Lee Metfords were upgraded, and the sights re-calibrated because the British adopted the Mark 7 Cartridge, a pointed, spitzer type, that gave flatter trajectory. This was done before, and during the early years of WWI.
The Lee-Enfield thus had a manufacturing life in the Military from 1895 to 1902, a span of 7 years. Of course, it is possible that stocks were on hand to complete more rifles, and private purchases could have been made later due to the factories concentrating on producing SMLEs. It could also be a Lee-Metford that has been factory rebarreled. If so, it is a later Lee Metford design, with the safety catch on the bolt.