That is a No. 1 Mk I (or No. 1 Mk I***) rifle which was obsoleted for army use by the time WWI started. The No. 1 Mk I*** rifles were used mostly by the Royal Navy. It is possible that some units may have been equipped with them, but by and large, the majority of Tommys were issued No. 1 Mk III rifles. I should mention that the original No. 1 Mk I rifles were simply referred to as SMLE Mk 1. The use of "No." did not take place until the revision of the numbering system in the 1920s.
The slidiing charger bridge was originally a modification to long Lee-Enfield rifles of various patterns, but the designation of Mk II.P was eventually established for all such converted guns. It was carried over as a feature of the first Short Lee-Enfield (SMLE) No. 1 Mk I, first introduced in 1902.
The subsequent No. 1 Mk III's and III*'s were manufactured with a one piece charger bridge obviating the necessity for the sliding piece on the bolt head, and many No.1 Mk I's were later converted to the one piece charger bridge (Mk III standards).
While the No.1 Mk I's are somewhat rare themselves, the trench periscope is quite a rare item.
The fact that the museum specimen is wire wrapped indicates that it was used for grenade launching purposes.
All in all, a very interesting display.