Many non-conformist church members had to be buried in unconsecrated ground, as their religion was not recognised as such by the established church.
A non-conformist church is any church that does not conform to the 39 Articles of Religion published in the Anglican Book of Common Prayer or does not recognize the Church of England as its parent body. This includes all the variants of Methodism, the Catholic Church (after the restoration of the Crown), Jewish synagogues, Baptists, Quakers, etc. Members of these faiths were also called "dissenters".
The Forest had many non-conformists so I suspect this is where your answer lies.
To be buried "under Burial Laws Amendment Act 1880" means that the person buried was a non-conformist; the burial service was performed by a Non-Conformist minister, but in a Church of England church as the burial was going to take place in the churchyard (consecrated ground).