Update: We're tracking a couple of reports that Microsoft EMET might cause the same issue, as it's the same type of application-restriction tool for system admins. Please be aware if you're running this tool, Microsoft's own literature says this is for enterprise systems only and there is no version out yet which actually supports Windows 10, with the most recent release being June 2014.
The Issue
Some users are reporting an issue where Studio fails to start correctly and instead they recieve a Windows crash message for the Updater. Checking the trace.log shows the following:
Unhandled Exception - System.ComponentModel.Win32Exception: This program is blocked by group policy. For more information, contact your system administrator
at System.Diagnostics.Process.StartWithShellExecuteEx(ProcessStartInfo startInfo)
at System.Diagnostics.Process.Start()
Alternatively, the Updater might work, but Studio itself crashes before showing the Welcome screen. In this case, your trace.log might be fine, but you typically do not have a traceIDE.log in the folder "%appdata%\GameMaker-Studio\" (this depends on how the blocking tool operates, but most reports we have had so far there has been no log we could refer to). If Studio has been able to start and has closed for a GameMaker-side issue there will be a traceIDE.log file written out, so if this is missing then Studio was not able to launch at all.
This is Windows itself or a third-party tool installed on your system blocking the call to run Studio and its updater and is something you will need to address on your PC. In order to fix this issue you will need to open the Group Policy editor normally used by system admins to restrict users' ability to install or control software capabilities on their machines. Please note you will likely need administrator privileges on your account in order to do this.
If you are in a networked/business/educational environment you may find that your account cannot access this tool or gpedit has been "overruled" by a different software suite anyway. You should check with your IT admin before proceeding with this guide.
CryptoLocker Prevention
One tool we have had reports of causing this issue is CryptoLocker Prevention (and by extension the CryptoLocker Prevention Kit available to some Windows users). This tool is used to block malware and other OS exploits by disabling executable files from running if they are in certain areas on your hard disk. One of these areas is %appdata%, which is where GameMaker: Studio is actually installed. You can amend the rules that CryptoLocker Prevention obeys via the following steps:
1) Click Start > Run (or Windows key and "R" at the same time) to get the Run dialogue, then type in gpedit.msc as shown below and click OK.

Once the editor is open, expand Conputer Configuration > Windows Settings > Security Settings > Software Restriction Policies to reach the below screen. Your machine might have existing rules on this screen, mine does not.

Right-click the Software Restriction Policies folder in the tree and choose "New ...". This will add a couple of premade rules into the main window pane on the right and a couple of subfolders into the tree. Click on the new Additional Rules subfolder to change the main panel. Now right-click in some empty space in the main panel and choose "New Path Rule". This will open the following popup:

In the Path field, add "%appdata%\GameMaker-Studio" (without the quotes) and in the Security level drop-down, ensure it is set to "Unrestricted" Your screen should look like this:

Now click OK to apply this rule and then close down the group policy editor. Try starting GameMaker: Studio again, and hopefully this will allow the updater and the IDE to start working.
Depending on your machine setup, you might need to reboot in order to have this rule apply, so if Studio still fails to start, please try restarting the PC next. If this still fails or gives you a different error, we'd recommend temporarily disabling the protection in your custom tool and running Studio to confirm that it will now start. If not, it's possible that the install has become corrupted and you will need to reinstall and try launching Studio again while the tool is disabled.