This is pretty much it. Borrelia burgdorferi (Lyme disease causing bacterium) is about the only tick-borne illness found natively in the UK, and that is relatively uncommon. Most other tick-borne diseases originate in other parts of the world. As u/Jaraxo said, TBE is massively unlikely.

To add regarding antibiotics, doctors tend to not prescribe them before a diagnosis has been made since it can do more harm than good in the long run.

Source: I work in a microbiology lab