If files already have been downloaded then you need to remove the files in the updates and updates\0 folder.
You can delete active-update.xml and updates.xml as well if present.

C:\Users\<user>\AppData\Local\Mozilla\Firefox\Mozilla Firefox\updates
(%LOCALAPPDATA%\Mozilla\Firefox\Mozilla Firefox)

If there are problems with updating then best is to download the full version and uninstall the currently installed version.

Download a fresh Firefox copy and save the file to the desktop.

• Firefox 6.0.x: http://www.mozilla.com/en-US/firefox/all.html

• Uninstall your current Firefox version.
• Do not remove personal data when you uninstall the current version.

Remove the Firefox program folder before installing that newly downloaded copy of the Firefox installer.

• It is important to delete the Firefox program folder to remove all the files and make sure that there are no problems with files that were leftover after uninstalling.

Your bookmarks and other profile data are stored elsewhere in the Firefox Profile Folder and won't be affected by a reinstall, but make sure that you do not select to remove personal data if you uninstall Firefox.

I have done as instructed. The Help >About Firefox screen still displays 'Apply Update' rather than 'Check for updates'.

Did you find the updates folder?

If you see an "Apply Update" button then that means that files are already downloaded and in that case Firefox won't do another check.

Is the current update channel set to Release (Help > About) ?

I have reinstalled Firefox from scratch, and deleted the updates files. After installing an old version it updates correctly, but then does not check for updates. I have found that changing the selected option for When updates are found from:

Ask me what I want to to do

to

Automatically download and install the update

fixes the problem

This suggests that there is a problem with the 'Ask me what I want to do' feature.