Grass Snakes eat a wide variety of things: frogs, toads, fish, newts, tadpoles, sometimes small mammals, slugs, nestling birds, other snakes, even salamanders. Young feed mainly on tadpoles and invertebrates.
So, they can eat mammals, birds, slugs and reptiles, but if amphibians (and fish) are available, they'll eat those instead.
Also, yes, the Aesculapian snake can climb very well and is known to eat birds. One has even been documented of constricting a bat, but didn't eat it. The people that saw this think it didn't eat it because they disturbed it.
French snakes that would eat birds (and eggs):
Western Whip Snake (Hierophis viridiflavus), Ladder snake (Rhinechis scalaris), Aesculapian Snake (Zamenis longissimus), Grass Snake (Natrix natrix), Asp Viper (Vipera Aspis), Adder (Vipera berus), Seoane's Viper (Vipera seoanei) and Montpellier snake (Malpolon monspessulanus)