Rooms 101 to 128 do afford a lovely view of Tavistock Square, and their balconies are a very nice feature. But regrettably, it is not allowed to go out onto the balcony.
The balconies (and their railings) on the first floor are not structurally certified to be able to support your weight. It is unsafe to stand or sit on the balcony.
Your Hall Handbook says this in section 14.1:
It is strictly forbidden – and extremely dangerous – for residents or their guests to gain access to, or attempt to gain access to, the roof of the main building, the roof of the dining hall, or any ledges or parapets on the Hall exterior, including first-floor balconies.