The Ticket Sales Office of HM Tower of London is listed in BT Directory Enquiries as 0870-756.7070. I could not find a geographical number for this office.

Instead of going diret to the ticket office, why not use this site http://www.2for1entry.co.uk/  and find the London Tower link. It's often on the right hand side.
The title is self explanatory, and it can be booked on the web.
Also, the Tower want to charge you £15 per adult, with a transaction fee (http://www.hrp.org.uk/webcode/content.asp?ID=34), but the site will do it for £14.50. For 2 adults you could save £15.50.

Or depending on what else you want to do that day, why not try /www.londonpass.com/. You can get your travel and loads of discounts or freebies, including a 50 minute Thames cruise.

The numbers are:

The Tower of London    0870 756 7070   01865 324121
Hampton Court Palace  0870 753 7777   01865 324120
Kensington Palace        0870 751 5180   01865 324122

Taken from their website here.

Please remember to dial 141 to withhold your number before dialling the geographical numbers listed above.

Many thanks for the excellent tips.