Re Sky, this solution would not enable you to record freeview channels I believe as the record function is only enabled when subscribing
Re installing an antenna, does your lease allow this, some blocks of flats don't allow them, but there is nno fixed rule to my knowledge.
Does the flat have no communal aerial feed?, If not then possibly an outdoor aerial would be possible and would be the preferred solution if you can mount one and access the transmitter (a ground floor flat might mean you can't get to the transmitter as its on the wrong side of the building etc
Indoor aerials can work, but depends on how strong your signal is, might be a case of try and see, no idea sorry about best of these
An indoor aerial might work, but only if you're in a really strong signal area. Be aware that the YouView boxes supplied by BT need a reasonably strong signal to work reliably - more so than many TVs.
I used to live in a top floor flat, and got away with a full-size aerial attached to a shelf up near the ceiling (above head height). But it looked really ugly!