I'm a research scholar in JNU (Jawaharlal Nehru University, New Delhi). Now I'm doing a paper in International Political Economy of Windows and Ubuntu. These are two similar products that use different strategies. I need data related the circulation of Ubuntu: how many users, which countries have more Ubuntu users, etc. I have been searching for this for a long time.
You may be able to get some of the data (or at least an estimate) from Canonical itself. Here is their contact page.
I recommend first contacting Canonical for this specific question. If anybody knows, they would.
Canonical Contact Page - http://www.canonical.com/about-canonical/contact
Ubuntu Contact Page - http://www.ubuntu.com/contact-us
With those two out of the way, to give an idea (Not an actual estimation) you have several sites that did some work (Most of which only compare Linux, Not only Ubuntu):
Pingdom - http://royal.pingdom.com/2011/05/12/the-top-20-strongholds-for-desktop-linux/ (Which I can talk about Venezuela. About 1 out of 4 people related to software know what Ubuntu is and 2 out of 5 have installed or tried it. You can go to any college and rapidly gather about 100 people that have used or seen Ubuntu in action.)
Wikipedia - And I quote:
Chris Kenyon, vice president for OEM at Canonical Ltd., indicates that because of a lack of registration, any numbers provided for Ubuntu distributions should be treated as estimates. Nevertheless, a number of estimates and the results of surveys have been provided. In January 2009, The New York Times reported that Ubuntu had over ten million users. In June 2009 ZDNet reported, "Worldwide, there are 13 million active Ubuntu users with use growing faster than any other distribution.", though Kenyon provided a more conservative estimate of 12 million users in April 2010. In fall 2011 Canonical estimated that Ubuntu holds a global usage of more than 20 million users. According to an independent survey by W3Techs.com Ubuntu is fourth most popular on web servers and rapidly increasing in popularity as web server, according to queries by Built With.
OMGUbuntu - http://www.omgubuntu.co.uk/2009/03/which-country-loves-ubuntu-most/ Which mentions which countries love/search Linux the most. I actually am surprised not to see in the graph Venezuela, even though the government promotes a self-made derivative of Debian, Canaima, you can bet there are more users that like Ubuntu and have it installed.
Ubuntu Counter - http://ubuntucounter.geekosophical.net/ (Not updated. I do not see 11.04 or 11.10 in there)
Linux Counter - https://linuxcounter.net/
But even using all of this sites almost as a base for some research there are other effects that affect how the trends go. For example, the amount of Ubuntu users lowered when Unity was used as default in Ubuntu 11.04. This has since started climbing but not as fast as one thought. 11.10 of course has a bit more users since Unity is more stable and reliable but still, the amount of Ubuntu users compared for example with 10.10 is not the same.
Add to that that Ubuntu's little son is growing up (Am talking about Mint) and people start looking somewhere else (Although, in my opinion there is not much to look at in terms of performance of ease of use, taking aside the default stuff that gets installed)
Not to mention users that install Ubuntu in multiple PCs but only mention they have installed Ubuntu in one (For stats and such).
Another point is the changes that might come in 12.04 and could affect/decide how the trend for Ubuntu will go. Basically right now is a very difficult moment to check out the amount of Ubuntu users.
Lastly a great idea for Ubuntu and to track how the progress of Ubuntu users is going is to add when installing Ubuntu an option to send (Similar to Linux Counter) some info about the PC and version of Ubuntu used. This by default. This way, for future references an easier study could be done to know the amount of users, hardware compatibility, etc.. (Similar to http://friendly.ubuntu.com/ but with a user counter) or verify which IPs download from the main repositories and for what version of Ubuntu.
So, all in all you can find some information for the research but it will be tough to give a very good estimate.
In case Canonical does not release the information for some reason, there are indirect means by which you can estimate a figure. The most obvious would be looking at countrywise search trends
Fedora (another linux distribution) maintains some statistics. To quote the page:
Currently, there is no reliable way to determine the total number of Linux users, or even count the total number of users of any Linux distribution which does not have a mandatory per user registration process. Anyone who tells you otherwise may be misinformed, dishonest, or trying to sell you something.
If canonical stores hashes of IPs hitting its update sites, then I guess you can have a pretty close number from them. One blogger estimated the figure to be around 24 million (in 2009).
I doubt anyone can give correct figure!!
Whatever figure you will get actual deployment will be 50% more than it as not every deployment gets tracked.