A few observations from points raised above:
It's most likely the pilot would not have been using a bone dome. Corporate pilots in the A109 do not wear them, at least I've never seen one doing so.
TCAS may have been fitted (many A109s have it), but it is probably irrelevant to the accident investigation because it is only of use for indicating other aircraft fitted with a transponder, not against ground based obstructions.
I would think it unlikely that the aircraft had a serious icing problem that would cause it to lose altitude. However, the plastic windscreens of A109s are not normally of the directly heated type, although electrically heated glass screens are an option on some marks, so they rely on the main aircraft cabin/cockpit heater to keep them clear. Therefore, misting or light icing on the screens, causing a reduction in the view of the pilot can't be totally ruled out.
Regarding the route being flown, i.e. inbound to Battersea heliport via Vauxhall Bridge. The accident occurred right at the place where the pilot is normally directed to change frequency to Battersea, i.e. joining H4 on the river and about to enter the Battersea ATZ. The Battersea controller does not have radar and would probably not be able to see an aircraft joining the river at Vauxhall bridge, even in CAVOK conditions, due to line of sight limitations.
An aircraft heading south, down the east side of the LHR CTR (as often occurs, to keep it out of Class A airspace, and inside London City zone), would be heading almost straight for the tower and the pilot would need to turn almost 90 degrees right to join heliroute H4 on reaching the river. The river is about 250 metres wide. The tower lies just inland of the south bank.
Some folks continue to think he would have been flying east at this stage. Unless the pilot was completed disorientated, or turning east to avoid deteriorating weather, it's unlikely. Battersea Heliport lies to the west of that point.
I did hear a TV "eye witness" report of a helicopter flying east, but the times don't add up and the witness said that was at 07:37. It's possible that another helicopter flew the route eastbound prior to the arrival of the one in question.
The AAIB will consider all these factors and because it occurred in controlled airspace it will be very well documented, probably far more comprehensively so than many aircraft accidents.
Finally and unfortunately, the UK building planning process/regulation doesn't seem to take into account the possible effect on helicopters that have to fly near them on long established routes.