Ah....... for a start, if I understand your description annie........
The felt tiles are folded over, then the plastic board (I'm guessing fascia board) fixed through the hanging down bit of the tiles, then the guttering?
If that's right, that could be the problem.
The fascia should be UNDER the tiles. These tiles should drip into the guttering.
I can't think how anyone could ever come up with the arrangement you have at the moment.
You need 2 things done.......
1 stop water getting in..... Have a sensible flat roof guy "torch" a new strip of felt around the perimeter. Bear in mind that he may advise going over the whole area if it's coming to the end of its useful life.... I can't tell from here.
2 For a pukka job, lay "Promenade" tiles all over. Have a look here.......
http://www.eurodec.uk.com/index.html
Certainly ring this company and talk through the technical side, to put your mind at rest.