The coldest wettest holiday in Europe I have ever endured was March in London( March 17ths, St Patricks day was lots of fun in pubs) . It was windy, rainy, and it sleeted( rain snow mix). I had a wool coat and a scarf.. but had no gloves or a hat, had to buy them after one day wearing my scarf wrapped around my head like a freak.
James, one problem you have is you are going to experience full range of weather from north( London in March) to south ( greece in April).
If it was me I would wear my heaviest items, that would be a decent lined raincoat, and a sweater,, and bring a few turtle necks or long sleeved shirts to to layer.
Good comfy walking shoes.Boots would be a pain to haul around in Greece I think,, and you won't need them there likely .