When the floorboards came up in the attic of Westminster Abbey, centuries of rubbish and thousands of pieces of fragile treasure were revealed: the sackloads of dust were sieved to make sure nothing was missed. The finds included a wooden Dutch clog, a clay pipe and 31,000 fragments of stained glass, some dating back to the 13th-century construction of the present church.

Some of the fragments came from windows blown out in the Blitz, some from the glass-smashing fury of the Reformation iconoclasts. All have been saved and carefully cleaned, and thousands are being reassembled into windows for a glass bridge linking a new tower, which will bring the general public for the first time into a hidden world.

Prince Charles, the patron of the £23m project, came on Wednesday morning to lay a foundation stone for the tower and take the builder’s lift up 70 ft (21 metres) to the triforium, the attic space that will become a museum named the Queen’s Diamond Jubilee Galleries in honour of his mother.

Visitors will be able to use either a lift or climb a winding staircase up to the oak, stone and glass tower: the slog up the stairs will be rewarded by a startling view of the abbey’s roofscape, bristling with statues and heraldic animals previously visible only to roofers and pigeons. Although it is the most dramatic intervention in the fabric of the abbey since towers were added to the western facade by Nicholas Hawksmoor in 1745, the new tower will be almost invisible behind a flying buttress and a towering plane tree. The planning application was passed without a vote by Westminster council, because there was not a single objection to it.

A statue is covered during restoration work at Westminster Abbey. Photograph: Dan Kitwood/Getty Images

The attics are so grand, with handsome carved stone decorations that would never be visible from the church floor, that the dean, the Very Rev John Hall, thinks they were probably originally intended as extra chapels for priests paid to say masses for the souls of the wealthy dead, but never used for anything except storage. New grilles and handrails will allow visitors to step right to the edge of the open arches, for dizzying views down to the 13th-century Cosmati pavement, and the royal tombs far below.

The space, due to open in 2018, will be used to display some of the abbey’s treasures, including the battered but magnificent Westminster Retable, a painted, carved, and gemstone-studded screen believed to have been made in the 13th century for the High Altar. It contains star-shaped motifs, formed by rotating one square on another, which the architect Ptolemy Dean has adopted as the floor plan of his new tower.

The new museum will complete the present development programme for the abbey but the dean still yearns to take on one more project: the central tower and spire that Hawksmoor proposed and began work on. Work stopped for the coronation of George II and never started again. “It won’t happen in my day,” Dean said wistfully, “if I were going to live for ever, I’d take it on.”