You asked

What happens if a school doesn't respond to a complaint of bullying?

Because bullying is such a serious issue that has received much awareness of late, it is highly improbable that the school won’t respond t a complaint. However, if you have approached the school with a complaint regarding bullying and you feel it hasn’t been responded to in the appropriate manner then you can take the issue further.
First, it’s important to understand that dealing with cases of bullying can take time and it may be the case that the school are taking the necessary measures but aren’t seeing the results.
However, if you believe that the school have taken the instance of bullying as far as they are willing to and your child is still being bullied you may need to submit your complaint in writing to the chairman of the Board of management for the school.
In your letter, you should state clearly the exact issue that you have, the steps that have been taken so far to resolve the issue and what you would like done to stop the bullying.
You should keep a record of this as you may be required to attend a meeting regarding the issue.

More questions

It's important to speak to your child about bullying to help them understand what behaviour is and isn't acceptable.
Recently, there has been an increase in the amount of children being sent abusive text messages.
There are many ways that you can help your child make friends at school. 
It's an unfortunate fact of life but kids do tease each other. It's important to teach your child how to deal with being teased. 
Children who bully want to exert control over their lives-even if it's just making someone else's life miserable.
This can be a difficult realisation to come to but there are some signs to watch out for which could indicate that your child is a bully. 
Parents often fear that if they report that their child is being bullied to the school that nothing will be done.
Your child's school should have a very clear policy on how to deal with bullying.
In recent times, cyber bullying has become all too common. 
There are lots of things that you as a parent can do to help prevent and reduce the chances of your child being bullied.