Black and white warblers?

Hi all, my first post, I may be mistaken but I just saw 2 black and white birds in my garden through the back door not 3 feet away on a rose bush, white stripe down the centre of the head about the size of a robin and black and white all over. The black and white warbler is native to North America, but I’ve read that on rare occasions it is seen in the UK? They behaved quite like a nuthatch would, moving up and down looking for insects. Unfortunately I was unable to get a picture, but that’s the only bird I can find online that fits the 2 I saw. Any input appreciated.