Insulating quarry tile window sills

[Oly]

My first question posted...  my late '60's house has quarry tile interior window sills, they look in keeping but are absolutely freezing. They are bridging the cavity and must be working like the opposite of a radiator, sucking warmth out of the building. I have 13 of these window sills in all, two of them are over 2.5m long and they are mostly 250-280 mm deep.

 

There's a fair bit of space above them (an inch) to fit something over without spoiling the window profile. I really don't want to use plastics, but thinking of timber with a square edge of bullnose added, perhaps trapping a layer of insulation (thin insulation board?) underneath.

 

Cost is an issue, so perhaps painted moisture resistant MDF plus a hardwood edge/bullnose. 

 

I'm wary of removing the tiles completely, but perhaps that's the best way of sorting this.

 

I'm also wondering about adding something like a thin insulation board to the reveals (I doubt the cavity closures are much good) that could be painted (and then frame the window opening entirely with timber square edge).

 

I'd love to hear if anyone's tackled this issue and the solutions.