Mice probably

If tghe whole of the plant has been eaten it might be rabbits. Do you get rabbits in London?

Try putting slug pellets in the bottle with the sunflowers. And it might help to use one of those multi purpose bug killer types of granules. My backyard is a converted field and I use a multi-type to catch a wide variety of harmful bugs. But the bottomless bottle is good against mice, rabbits and pidgeons; and it protects your pets getting curious.

Bulbaholic you made me laugh! Yes we have rabbits in London! My daughter has 2 rabbits one in doors as he is eperletic and the other is in a hutch outside and goes in the rabbit run regularly. So its not them.
As Wylieinthea suggests (nice name by the way) I guess it could be pigeons. multi purpose bug killer sounds good but my garden is organic. A slug pub comes to mind. thanks guys

I planted sunflower seeds one year and the mice ate every one !

I think Bulbaholic meant wild rabbits, Great, not domesticated ones, lol! And yes, Bulbaholic, we certainly do have rabbits - but they usually restrict themselves to golf courses and the like.

If the sunflowers are covered with 2 litre bottles then I don't see how rabbits wild or domesticated could eat them... I still reckon it is mice.

or voles, they are so tiny and do move through the soil

Thanks, Bamboo, yes I did mean wild and coming into gardens. A bit like urban foxes.

Perhaps I should do some detective work and put cameras on my sun flowers lol and see who the culprits or culprits are. No not really