The answer seems to be that it was not explicitly prohibited before 1880; the key phrase is "without the conformist rites of that Church", so before 1880 they simply had to choose to accept the Anglican burial service or get buried elsewhere. Per these notes:
Thus common law conferred on the parishioner the right both to burial and to a burial service. The only distinction was that the work was divided between two different ecclesiastical persons. However, it was not possible to choose burial but reject a burial service (or vice versa). Canon 68 of 1603 made clear that burial and the burial service had to be performed together. [...]
The inseparability of the burial from the Anglican burial rite caused friction between the Established Church and other denominations, who wished to bury their departed adherents according to their own rites. [...]
The Burial Laws Amendment Act 1880 eased the sectarian tensions by permitting burial of non-Anglicans in a churchyard ‘either without any religious service, or with [a] Christian and orderly service at the grave’ (s.6).
So someone could be buried in an Anglican churchyard pre-1880 as long as they were willing to accept it being done with an Anglican burial service, and the Anglican church seems to have been required to carry that out.
This note (from Northamptonshire) suggests it varied a bit by denomination - Quakers and Baptists might be more likely to have their own burial grounds than Methodists - but presumably that also varied depending on how large or wealthy the specific local nonconformist communities were, as well as on more individual factors like how committed the deceased (or their family) were. Some people presumably minded more than others!
