There are a few compact cameras with optical zooms that do 1080p/30 for under £120 if you search reputable places like Warehouse Express or Park Cameras. They seem to be badge engineered. The Kodak's have a "traditional" small sensor (some of the others are really tiny) and you can pick between a small 28-140 or a less compact superzoom. Image quality won't be up there with the DSLRs or the latest mirrorless cameras.
Or London Camera Exchange (not just London) have a second hand Sony WX300 that will do 1080p/60, is pocketable and has a 28-500 lens, and you'll have budget left over for a couple of aftermarket spare batteries and filters.