Should my cat be eating city Pigeons?

My cats used to live in the countryside with me. As a result, they became very skillful at hunting - even with bells on their collars. However, time has ped, I now live in the suburbs of Manchester, my cats are getting old and the weather has been quite hot recently. Nonetheless, this has not deterred one of them from going on, what can only be described as, an indiscriminate pigeon killing rampage – chicks and all. After killing a pigeon, my cat will rip it open and eat as much of it as he can. Is there any chance my cat could contract a disease from eating one of these pigeons?