Found vintage Elgin pocket watch with engravings

I found a vintage Elgin pocket watch this afternoon around 2:15pm by the Palace Pier on the Brighton seafront. It's a gold-tone case, about 4 cm across, with a white dial and black Roman numerals. The back is engraved with tiny initials 'A.J.' and there’s a delicate ornate pattern that catches the light. The crystal has a small crack near 3 o'clock, and the attached chain has a short, worn link, like it’s been carried in a pocket for decades. This is clearly a collector's item, and I’d really like to get it back to its owner—it felt special seeing it gleam in the sun. If anyone recognizes it, please get in touch.