It is probably a Smith & Wesson Military & Police (later called the Model 10). From your description it will be a .38 S&W not a .38 Spl. If it is an ex-police gun they almost always have some stampings and or markings indicating the police agency. Early RCMP revolvers, like the one you have, usually had an MP stamped on the wooden grips or frame or both. Because the RCMP took over from a number of Provincal Police Depts some revolvers could have markings from a number of agencies. One old M&P .38 S&W that I examined had marks from three agencies, RCMP (MP), Saskatchewan Prov. Police (S.P.P.) and British Columbia Prov. Police (BCP). I doubt it is an ex-police revolver if it does not have some markings indicating the Police agency. Look closely to find the identification marks.
I hope this helps.