Depends exactly how you're defining "iconic". Some animals are pretty special but are so rare or localised to particular areas that you could live here your whole life and never see one (e.g. red squirrel, pine marten, otter), whereas other animals are pretty "iconic" of the UK but are so common they're not exciting (e.g. pigeons). Also the animals that we'll pick will depend on which parts of the country we've lived in, as some animals are very iconic in certain areas but rarely seen in others.

For me (I live in Leeds, and have always lived around the north of England) I'd say it would be foxes (yes, they're common but still feel sort of special, unlike pigeons!), deer (red or roe), hedgehogs, red kites (this one is probably going to be localised, some areas have none while they're now widespread in others after a reintroduction programme), and maybe badgers (although you're much less likely to actually see a badger unless you're specifically setting out to find one). Or perhaps replace badgers with robins or blackbirds, both iconic British garden birds, but probably less exciting than a badger!