I imagine that the top, flatter section is also tile-hung? This shouldn't normally be a problem since any missing tiles should be readily apparent.
Assuming all the tiles are in place, there are two main possibilities. Perhaps there are one or two cracked tiles. If so, then they are easily replaced.
The other possibility is that, because the pitch is very slow here, strong winds could be blowing rain up under the tiles. In that case, it's a simple job to remove the tiles and battens... then re-fit the tiles with new battens... over new breathable felt.
As for the steep section, I agree with Togo about the vertical window soakers. It all depends on exactly where the water is coming in.