In short the answer is yes you can.
Solid roof conservatory planning permission.
Since the changes in 2008 with respect to planning there is no difference between a single storey solid roof extension a conservatory an orangery or a tiled slate roof garden room.
It used to be that conservatories had to have a certain percentage of their roofing be translucent in order for them to be exempt from planning permission but changes to building regulations now means that you may not need planning permission for a solid roof.
Prior to their changing to be exempt from planning permission conservatories had to have 75 of the roof glazed.
At the risk of oversimplifying the answer to do you need planning permission for a conservatory roof is no.
Planning permission is in short formal approval for the construction of any building structure or extension.
If you have any doubts as to whether or not your conservatory needs planning permission you should contact local building authorities immediately.
Planning regulations were changed to allow conservatories to have a tiled solid roof back in 2010.
Planning permission is the permission to erect or extend a conservatory orangery or garden room.
In 2010 the planning regulations changed to allow conservatories to have a solid roof without the need for planning permission.
So the short answer to the question of whether or not you need planning permission to install a tiled roof on your conservatory is yes.
As mentioned planning permission is a legal requirement which cannot be ignored.