Ok... This is what happens. Assuming you have set Facebook to remember your login, when you click the 'F' link from Mumsnet to Facebook, a 'Share' box comes up. If you then click 'share' on that share box, you will make a link on your Facebook profile to that particular Mumsnet discussion (thread).
A link to the thread will appear on your Facebook wall but it won't tell anyone who you are or make your Mumsnet username known (although your friends may guess at your identity if you make it obvious in the thread).
If you can't see the link to the thread on your profile, you probably didn't click 'share' on the share box so the link was probably never made.
It is /technically/ possible that Mumsnet's techy staff could perhaps follow this link to see what your Facebook profile is - but very they probably won't. Unless you have a stalker who happens to work for Mumsnet, I would not be concerned.
Bear in mind the admin of ANY website you visit can see your IP address, work out your general geographic location and see what search terms you used to find their website and/or the websites which took you to their website, as well as the website you visit directly AFTER visiting their website. Plus more info. No, there isn't a lot of privacy on the internet, but generally the people who see all this info about you aren't the people who know you personally.