It would not normally be possible to fly a free balloon within Controlled Airspace. However, in the past, such flights have been approved very early on summer mornings when the wind would carry the balloon away from main traffic patterns. One balloon, I recall, had a modern transponder on board so we could watch it.
Airships are fundamentally different to balloons as they can be steered!
You'd be best seeking proper advice from ATC Ops at Heathrow.
I'm sure that someone more clued up will respond...