I currently have a webpage which I know has at least 10 backlinks, but only one of these actually shows up in Google Webmaster Tools. This same webpage is also ranking much lower than I expected it to. So have these backlinks not been taken into account by Google yet? And will they be added to Webmaster Tools as soon as Google notices them?
There's a few things to take into consideration here:
Google may not know about all of those backlinks. The pages they are on may not be crawled yet, may be blocked by robots.txt, or may be nofollow. Until Google finds and indexes them they carry no weight.
Google may consider some of those links as low quality. If those links are part of a link exchange, link scheme, or any other means that Google considers low quality, they may not be counted as a link/reported in GWT.
Google may know about those links and just not reported them in GWT yet. There is no known timeline for how often Google updates GWT so it may simply be a case of the links are counting towards your rankings and just isn't reported in GWT...yet.
Your expectations are not realistic. We see it here all the time. 10 backlinks is nothing in terms of SEO. Especially if they are not quality links. You will almost certainly need more links than that if you hope to see a page rank well in Google.