Newton Heath has never been part of Chadderton or Oldham. When Newton Heath formed, the town was not part of Manchester, it was Lancashire. Manchester at that time was pretty much what is now the City Centre and the towns in the immediate vicinity (Ardwick, Ancoats etc). Newton Heath incorporated Miles Platting, and Monsall at the time and was a large area.
Neither the Newton Heath nor St Marks were born within the boundaries of what was then known as Manchester, at the time of their formation.
Newton Heath and Clayton became part of Manchester Central in 1890. St Marks moved to Ardwick in 1887. Ardwick had been part of Manchester from 1838.
Therefore the early incarnation of City were playing in Manchester before any incarnation of United.
City have never left the actual city of Manchester from 1887.
United moved to Old Trafford in 1910. Old Trafford has never been in Manchester. So, the only time any incarnation of United played in Manchester was between 1890 and 1910.