This is a character piece by Walford Davies, often associated with his BBC broadcasts. It is distinguished by a quintessentially British sentimental charm and atmospheric harmonic language.
Focus on the balance of the melody in the middle register against simple chordal accompaniment.
Comparable difficulty and character — good candidates for what to learn next.
Big Ben Looks On is rated Henle level 3 of 9 — late beginner repertoire, roughly ABRSM grades 4–5. In the RCM system it corresponds to about level 3.
You should be comfortable at Henle level 3 — meaning you can already play level 2–3 pieces cleanly. Learning it one level early is possible as a stretch piece, but more than that usually leads to months of frustration.
A typical performance of Big Ben Looks On lasts about 3 minutes.
Big Ben Looks On is available as free public-domain sheet music on IMSLP — the score is linked directly from this page.
Pieces at a comparable difficulty and character include Barafostus Dreame, MB 62 by Thomas Tomkins, Bogurodzica by Adalbert of Prague, Barafostus Dreame by Thomas Tomkins — see the full similar-pieces list on this page.