Hi Robbie,
Yes the family stayed in the Toxteth area, I don't know when they arrived in Liverpool but William married in St Philips Liverpool to Ellen Parr 1858, their mum Martha was a widow and living with boarders in 1861, the son Edward died in I think 1858, he had taken over from his dad as a lock-keeper, I found out that he is buried in Overleigh cem in Chester. I am not sure when James Cooper Clarke died, I think his death reg could be just James or even just Cooper Clarke.
I have been researching the Clarke's for a number of years with the help of my dad, uncle and cousin. I thought that the boys may have been baptised in Chester, but I found out that they were baptised in Flintshire, I did find James and Martha's marriage in St Mary's Chester, luckily enough it is available to view on line on FindMyPast website, not sure if I can mention the full name of the site on here.
Most of the Clarke's are still in the Liverpool area, as I am too.
I did contact Chester RO for the info about William & Timothy's records at the Bluecoat school, but as yet I have not purchased the info.
Edwards details from Overleigh cem if you click on the highlighted surname.
http://archivedatabases.cheshire.gov.uk/cms/LacOverleighCemetery/results.aspx?Surname=clarke&Firstname=edward&GraveNumber=&YearOfDeathFrom=1856&YearOfDeathTo=1858&AgeRange=Regards
Jan