Well, I'm about the farthest thing from an Enfield expert myself.
From world.guns.ru:
"The Lee-Enfield series of rifles was born in 1895 as a marriage between the magazine and bolt action, designed by the J. P. Lee, and the new pattern of barrel rifling, designed at the Royal Small Arms Factory (RSAF) at Enfield. Originally known as Lee-Metford, this design was adopted by British army in 1888 and used a Metford pattern rifling with shallow groves, intended to be used with ammunition loaded with black powder. Introduction of the smokeless powders in the form of the Cordite showed that the Metford rifling was very short-living, so it was soon replaced with Enfield rifling, with 5 traditional land and grooves and left hand pitch."
"RSAF was set do design another rifle, patterned after the German Mauser, which also should be more suitable for mass production, than the SMLE. This rifle finally appeared in 1914 as an ".303 caliber Enfield Pattern 1914 rifle", or simply a P-14. .... The "No.3" was assigned to the P-14 rifle"
During WWI the US was contracted to produce No3's. Most of these were chambered in 30-06 and called 'M1917'. See
http://www.surplusrifle.com/m1917/index.asp
Bolt details for the No1/No4 can be found at
http://www.surplusrifle.com/m1917/index.asp . I also have two No4Mk1's, and can attest to their NOT being a CRF design.
The No3, being a Mauser knockoff, has a CRF design as shown in Figure 13 at
http://www.surplusrifle.com/m1917/bolt/hs.asp .
It is interesting to note that the development of the Mauser copy (the No3) did NOT result in the adoption of that rifle. The No1/No4 design proved sufficient after all, and the Brits never really found the need to switch to the Mauser design.