If I search "traditional English pie" I see images of something more similar to what I suggested than the image you linked.
I'm in the U.S., so the search results might be affected by my location.
edit: I tried on google.co.uk and I get more results similar to what you suggested.
Yeah I think that's google trickery. How strange though, I would have thought the search "traditional English pie" shouldn't/wouldn't be affected by the location.
Nope, nothing like that, I think. English pies use thicker pastry, are filled with gravy and never anything green. Something like this[1] gives you an idea. Perhaps it's the sort of equivalent, but the Italian I work with says there is nothing like them that he's ever had in Italy.
1. http://www.bbcgoodfood.com/recipes/3148692/dads-beef-mushroo...