Your rifle is a #4 Pattern High Velocity Sporting Rifle, built & marketed by B.S.A.Co. between 1902 & August 1914.
#1 Pattern Deluxe had a barrel rib, full coverage engraving, horn fore-end & grip cap, engraved steel trap butt-plate complete with brass oil bottle and pull-through & usually 5 round fish-belly magazine;
#1 Pattern had a barrel rib, 3/8th coverage engraving, horn fore-end & grip cap; engraved steel trap butt-plate complete with brass oil bottle and pull-through & usually 5 round fish-belly magazine
#2 Pattern had no barrel rib, partial coverage engraving, horn fore-end & grip cap, engraved steel trap butt-plate complete with brass oil bottle and pull-through & usually 5 round fish-belly magazine;
#3 Pattern had no barrel rib, no engraving, ebonite fore-end & grip cap, un-engraved steel trap butt-plate & usually 5 round fish-belly magazine;
#4 Pattern had no barrel rib, no engraving, no fore-end or grip cap, gutta percha butt-plate emboosed with the B.S.A.Co. logo of three stacked rifles & a 10 round standard military pattern magazine.
While virtually all of the British "Gunmakers" retailed these rifles, with their own markings, 95% were in fact made by Birmingham Small Arms Co., while the remaining 5% were made by London Small Arms Co. Ltd., who used the carbine actions exclusively for their sporters. Typically, these latter were supplied to the big London (Holland & Holland, J. Purdey, W.J. Jeffery, J. Rigby, Wm. Evans, J. Wilkes, H. Atkin, S.Grant, Lang& Scottish (Fraser, MacNaughton & Gibbs) gunmaking firms as well as to Army & Navy Co-operative Society Ltd. (who also had the B.S.A.Co. rifles made for them).
The Lee SPEED PATENT marked ones were so marked until the 14 year British patents expired. Thereafter, they were not required to be marked to acknowledge the patents or to pay royalties for their use.
Prior to B.S.A.Co. building the HVSR's themselves, starting in 1902, a number of makers to the trade used the commercially available B.S.A. & M. Co. or the later B.S.A.Co. Volunteer pattern rifle or carbine actions or barrelled actions to supply James Paris Lee's sporting rifles to the trade, partly because there was a dearth of bolt action sporting rifles extant & partly because of not wanting to pay Rigby's mark-up on the Mauser actions for which he was the exclusive importer at the time. The only other bolt action commonly used was the Steyr Mannlicher Model 1892-95 turnbolts in .256 Mannlicher (aka 6.5x53R) or .303 Flanged.
These rifles are my collecting passsion & I am in the midst of writing a book on them. I am also trying to determine which of mine to take to the Dark Continent on a plains game safari.