You could contact West Lothian local history library (Google for contact details) who might look at the relevant local papers for you. There is a brief report in the Glasgow Herald - this is typical of the amount of detail you would get on such incidents, and it may be that a local paper would not contain any more information:
Melancholy Accident - Four Men Drowned in a Coal Pit - Early on the morning of the 7th current, four miners, named Robert Russell, John Hamilton, John Hastie, and Joseph Aird, belonging to Bo'ness or vicinity, met their death by drowning in a coal and ironstone pit, near Kinneil Iron Works. While the men were engaged at work in the bottom coal workings, a sudden flow of water .came in upon them from some old workings, and instantly drowned them. Fortunately a lad named David Spence, who was at the bottom of the shaft, heard the rush of the water in time to make his escape to an ironstone working about fourteen feet farther up. He gave the alarm to six or seven workmen there, and he, along with two or three of those nearest the shaft, were taken up in safety. So rapid was the influx of water into the shaft that by the time the cage got down again for the remainder in the ironstone workings the water was up to their middle, and it was with great difficulty that they could get into the cage and be drawn up. It is believed that some months must elapse before the water can all be pumped out and the bodies of the four men got. - Courant. [Glasgow Herald 17 August 1857]
