Family of three looking for apart or house

Hi!
We are a couple with a one year old baby looking for an apartment to rent for 3 months, from the beginning of September until the beginning of December, but we are flexible with the dates. We've been living in Liverpool for the last 4 years, but lived in Manchester and London before. We work from home so are flexible with the location, as we want to see different cities. We have in mind Manchester, Leeds, Glasgow, Bristol, Birmingham, London or anywhere in the South of England. We both work full time as web developers.