Unfortunately the only way to find out if this is your William Mills death in 1881 is to buy the certificate and hope that the informant, address, occupation etc match with what you know.
I looked on Lancashire OPC and Manchester Burials but nothing there.
The GRO say this on their web site so it may be that the purchase of the cert is not necessary. However I haven't done this so don't know exactly how it works. A phone call perhaps?
"If you do not have the GRO index reference you may still apply online for any event which took place on or after 1st September 1837 to 6 months from the current date (for marriages this period is extended to 18 months). A 3 year search for the index reference will be carried out. You must supply the exact date of event if you have it. If you don't have the exact date enter 01/01/YEAR and we will search the specified year and one year either side."
Also don't forget that William and Harriet had sons John 1870/1 and James 1873/4 who are with their father and grandmother in 1881. and in 1871 John is with his parents.
Yes I do think that this family I found and posted the 1851/61/71/81 census records for are your William Mills family.
I am just not sure about Reuben being the son of William even though his birth cert says he is.