The pods are classed as outbuildings and so typically fall within your permitted development rights which means you can have one installed without planning permission. There are of course exceptions to this rule so the following may be helpful:
If you live in an area called 'designated land.' This might be a conservation area, but these usually have fewer strict rules than national parks. If your pod can't be seen from the road, you probably won't have any issues.
Your pod will be located at the front of your house. This is known as the "Principal Elevation," and it's common for houses to have their main front facing the road. If your pod will be positioned in front of your front door, it's generally a good idea to seek permission.
If you live in a listed property you will most likely need to apply for permission to hang your washing let alone have a pod installed!