On two different computers? I hastily installed the client onto my desktop and my desktop I've ordered has been dispatched quicker then I expected (tomorrow) I've already used up the beta key, but does any body know if I could still install it onto my desktop tomorrow?

One of the ZOS people posted:

Hello everyone! We’re thrilled by the excitement here over The Elder Scrolls Online beta test. To prevent beta-related threads from overwhelming the forum, let's please keep all related discussion in here. Additionally, a few reminders:

• The Elder Scrolls Online beta test is under NDA (non-disclosure agreement); you shouldn’t be discussing details of the test in public places such as these forums, social media, streaming video, et cetera. If you’ve gotten in, it’s OK to let people know that, but you can’t say anything else about the test.

• We have a private forum for the ESO Beta; the account information you need to log-in was sent with your beta invite.

• If you are having trouble, or otherwise need help; you can check for solutions on our http://help.elderscrollsonline.com/, search the help section of the ESO beta forums, submit a http://help.elderscrollsonline.com/app/ask, or call customer support (details can be found in your invite). It’s very difficult for us to help if you ask here or on our social media channels, and chances are any such requests would break NDA.

• If people post other threads about beta, or break the NDA; please let them know of their error politely and direct them to this thread. If content needs to be removed you should report it with the button found under every post.

We don't know the answer. Your best bet would be to open a support ticket.

I wouldn't think that would be an issue, but you should ask CS that question.