The locomotive (belived now on route) is the National Railway Museum's, London and South Western Railway, 4-4-0 No.563 (see
http://www.nrm.org.uk/OurCollection/LocomotivesAndRollingStock/CollectionItem.aspx?objid=1975-7000&pageNo=14).

More information can be gathered from a discussion at "National Preservation" the UK equivalent of "Railway Preservation News"
http://railways.national-preservation.com/showthread.php/30041-LSWR-T3-4-4-0-exported-to-Canada.

If the "Railway Children" is goes down as well on your side of the Atlantic as it has done on ours, I think that 563, could be away from home for some time! The London production, which featured Great Northern Railway, 4-2-2, No.1 (http://www.nrm.org.uk/OurCollection/LocomotivesAndRollingStock/CollectionItem.aspx?objid=1975-7014&cat=All&comp=Great_Northern_Railway&ipp=12) and the same carriage as used in the 1970's film of the same name, is belived to be be starting a second run later this year. Hence, one reason why, the Canadian production, is using different stock.

Am I understanding correctly, that this is a stage show featuring a full size steam locomotive and rolling stock? And it all gets moved from one venue to another?

This will certainly be something to see if that is true; especially given the rest of my family's involvment in theatre.