I thought in Ukrainian ц always had an implied soft sign after it. Is this an exception, or is the general rule just in my imagination?
There is no such rule. Ukrainian ц can be either hard or soft (unlike Russian ц, which is always hard).
Yep, the "virtual soft sign" would appear only with ці, ця and цю. Ца, цо, цу, це, ци are hard, and цє doesn't exist (at least I can't think of anything). Also цьо, but the soft sign is explicit here.
Piccadilly Circus is a 'Площа Пікаділі' (i.e. we use the same word for the Trafalgar Square 'Трафальга́рська пло́ща' and for Piccadilly Circus 'Пло́ща Пікаді́лі').