I AM sure you have a good chance. In many areas there are vacancies now, and this autumn many governors complete their term of service.
You do not say whether your local school is a church-aided school or a community (until recently called a county) school. In the former case your only route is through appointment by the providing church. The simplest thing is to write to the vicar and ask if they need volunteers. They have plenty of interest among church members, and would want to be satisfied that you are sympathetic to their ethos.
Your chances in a community school are much better, as they have representatives of the local education authority and co-opted governors - between two and four of each in a primary, and five of each in a large comprehensive.
Many LEAs have abandoned the tradition of getting governors through politcial parties and are happy to have volunteers wth something to offer. Some even advertise. If yours is one of these, write to the governor services department at your local county or borough education department with a brief curriculum vitae and a statement of your interest.
If you became an LEA appointee you would not be obliged to toe any particular line. But you should perhaps take a somewhat broader view than a teacher or parent-governor might, and be familiar with and prepared to explain the LEA's policies.
If you are co-opted you don't have that obligation, but perhaps might bring a distinctive point of view as a neighbour of the school on behalf of its wider community.
Regardless of who they represent or who appoints them, most governors understand that they are there first and foremost to consider the interests of the children.
If you want to be a co-opted governor you should write to the chair of governors at the school, again enclosing a note about yourself and asking to be considered when there is a vacancy. The other governors choose the co-optees as and when there are vacancies.