I visited the site of the Stock Exchange Tower on Wednesday 29th March 2006.
This is what I saw.

As well as these photographs I spoke with a site worker who told me the following:
There is a height limit for new buildings built close to St Paul's Cathedral so, although it would be cheaper to demolish the existing Tower and build a new one, this would not be allowed, and so the decision was made to renovate the Tower as happened to Brittanic House a few years ago. The renovation involves widening the Tower from the 5th floor up to the top floor, by 2 metres all round, by cantilevering out from the existing pillars, and then cladding the entire building in glass, as is the modern style. To the west of the Tower will be built an 11 floor area incorporating shopping and restaurants. The old Tower included two equipment floors part-way up and two more floors at the top. These will be replaced by two new equipment floors at the top, thereby releasing more office space overall.

1. View of the South side of the Tower across Threadneedle Street from the corner of the Royal Exchange,
showing the demolition of the low-rise Western part of the Tower.

2. View across Old Broad Street from the corner of Threadneedle Street,
showing the removal of the windows from the bottom of the main Tower.

3. View from the Eastern side across Old Broad Street,
showing how the lower part of the Tower has been gutted.

4. View from the corner of Throgmorton Street and Old Broad Street,
with the Post Office totally gone!

5. View from Throgmorton Street,
showing the exposed stairways.

6. View along Throgmorton Street,
with the Bank of England in the distance.

7. Sad sight.

8. View from Throgmorton Street, on the North side of the Tower,
showing the extent of the removal of the low-rise parts of the tower
(the darkened windows partway up are two floors of equipment rooms).

9. View from Throgmorton Street, showing the North and West sides of the Tower.
It still looks like the old Tower, with the old Nat West Tower to its left, although the
Nat West Tower is now called Tower 42 and I suspect the Stock Exchange Tower
won't be known as that after its refurbishment.

10. View from Throgmorton Street, showing the North and West sides of the Tower,
showing how much of the ground floor area has been demolished.

11. View from Throgmorton Street, showing the West sides of the Tower,
the two outcrops show mockups of the new glazed cladding which will envelop the new
tower, extending beyond the existing concrete pillars, and the scaffolding to its right
shows how the work will be carried out descending from the top down.