Hope you are feeling well? Do you have any symptoms? Here is a link of many things you can get from birds and these are not all from a splash or drinking the bird water by accident. A lot are acquired other ways, such as inhaling pigeon droppings, or a bite from a rat or other animal or getting it in a a open wound.
However, it is unlikely. If you don't have any symptoms after a week or two I wouldn't worry about it. Generally, if you were to get anything from a splash, it is rare but bird flu, Salmonella, and Mycobacterium avium . Bird flu would likely be a bad flu but you would know by now. Mycobacterium avium is more from a wound you would get not from a splash . and it is extremely rare, and usually not from the U.K They only treat the wound not oral antibiotics etc. Salmonella, you would have high fever and diarrhea. The disease you would get from rats or rodents would be from a bite not generally ingestion. Have you seen any rat droppings in the bird bath or animal droppings other than bird droppings? Did you get it in your eye? Eye infection is possible, but you would also know that by now. If not then I doubt you got anything from this.
Hope this helps.