Wikipedia has an article about the theatre:
St. Martin's Theatre is a West End theatre which has staged the production of The Mousetrap since March 1974, making it the longest continuous run of any show in the world.
The theatre is located on West Street, near Shaftesbury Avenue, in the West End of London. It was designed by W. G. R. Sprague as one of a pair of theatres, along with the Ambassadors Theatre, also on West Street. Richard Verney, 19th Baron Willoughby de Broke, together with B. A. (Bertie) Meyer, commissioned Sprague to design the theatre buildings. Although the Ambassadors opened in 1913, construction of St. Martin's was delayed by the outbreak of World War I. The theatre is still part-owned by the Willoughby de Broke family along with Stephen Waley-Cohen.
The first production at St. Martin's was the spectacular Edwardian musical comedy Houp La!, starring Gertie Millar, which opened on 23 November 1916. The producer was the impresario Charles B. Cochran, who took a 21-year lease on the new theatre.
Many famous British actors passed through St. Martin's. In April 1923 Basil Rathbone played Harry Domain in R.U.R. and in June 1927 Henry Daniell appeared there as Gregory Brown in Meet the Wife. Successes at the theatre included Hugh Williams's play (later a film) The Grass is Greener, John Mortimer's The Wrong Side of the Park and the 1970 thriller Sleuth.
After Cochran, Bertie Meyer ran the theatre intermittently until 1967, when his son R. A. (Ricky) Meyer became administrator for the next two decades. St. Martin's was Grade II listed by English Heritage in March 1973.
In March 1974 Agatha Christie's The Mousetrap transferred from the Ambassadors to St. Martin's, where it remains today, holding the record for the longest continuously running show in the world. It has exceeded 25,000 performances at St. Martin's.
As mentioned, the theatre is Grade II listed with the entry at the English heritage website telling us:
Theatre. 1916. By WGR Sprague as a companion to the Ambassadors (1913), West Street (qv), but for different owners. Built by Lenn Thornton and Co. Ashlar facade with banded rustication to ground floor.
EXTERIOR: 4 storeys. 5 bays with 10-window left return. Ground floor with round arch openings and wide central entrance surmounted by a plain C20 canopy. Giant engaged Ionic columns in antis with rusticated end pilasters rise through the 1st, 2nd and 3rd floors, supporting an entablature, with triglyphs in the frieze, and parapet. First floor windows have Tuscan pilasters supporting pediments, above shouldered architraves and keyblocks.
INTERIOR: 2-tier auditorium, in Classical style, of polished hardwood. Turned balusters to balcony fronts. Side walls of auditorium have large Doric engaged columns and pilasters on pedestals supporting bold entablature which continues over the rectangular proscenium arch. Glazed domed ceiling. Stage machinery: nearly complete example of Edwardian (1916) wooden stage, marred only by alterations made to accommodate dimmer room in the sub-stage space. Represents one of the final stages in the development of timber stage machinery in England and an exceptional survival.
HISTORICAL NOTE: "Architectural Review" of that time noted a change in style of decoration from the lavish showy palace to a traditional English domestic interior implying a more select audience.