If I buy an additional Alcatel speedtouch modem I will be able to access BT broadband from either of my computers - but wiil each of them be able to be online at the same time ?
I don't know the answer to your specific question, but if you set up a network between the computers they can share the same modem connection. The later versions of windows have an Internet Sharing option which makes it fairly easy to install.
Your first modem will hog the ADSL line and will become unavailable to the second modem. So, unless you get another ADSL line you will not be able to surf at the same time. Why get another modem if you can share one? smorodina's suggestion about ICS is spot on.