I had some problems in running Cortana on my PC/tablet updated to Windows 10 (even updating my Surface 3 pro that was in Italian!).

I found the following solution, that worked for me, in a thread of  Microsoft Community forum:

It is used UK as example.

1. Click SETTING and then get into TIME&LANGUAGE

2. Click COUNTRY OR REGION

3. Change place to United Kingdom and add  English(United Kingdom)  as your default language.

4. Change you speech language to English(United Kingdom) 

5. Click ‘Additional date, time & regional settings’

6. Change location to UK

7. (important step) After change location, do not close your page, just click Administrative and click Copy settings.

8. Tick all you can see and click OK

(VERY IMPORTANT)

If you see the language of Current user, Welcome screen and New user accounts are all English(United Kingdom), you will find Cortana work after you restart computer.

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However I did not succeed to have Cortana works choosing Display/Input/Format language => English (US) and maintaining my real Location => another supported Country (e.g. Italy).

In fact I would like to maintain the UI of Windows to English (being a developer) but my location is Italy (not United States as I set the Display/Input/Format settings).

Therefore, for the time being, while if I put all setting to Italian/Italy Cortana works, if I try to set the UI to English and I leave my location to the real one (Italy), Cortana stops working (“Cortana is not supported in the region and language you’ve selected”). Waiting an answer from the community …

NOTE: you can use Cortana also if you are not logged with your Microsoft user (e.g. a local account). However be careful if you are using your Microsoft account and you use it on more PCs: in fact in that case it is better to disable the Language preferences synchronization. If you do not do so, if you put a different Windows display language in your PCs, possibly at the next reboot something wrong could happen on some of your PCs because of that synch with the setting of your account saved and synchronized with the cloud. You could have the Windows language changed or only some other language setting … and possibly have Cortana not work anymore on a PC where it worked (i.e. the one you put region and language compliant, like Italy and Italian). In that case even the voice recognition can fail as it is shown in the following screenshots.

Voice recognition fails

Set to OFF the synch of the Language preferences

Clock, Language and region -> Language -> advanced settings

Clock, Language and region -> Language -> advanced settings – Apply language settings to the welcome screen and new user accounts

Clock, Language and region -> Language -> advanced settings – Apply language settings to the welcome screen and new user accounts -> Welcome screen and new user accounts settings

Clock, Language and region -> Language -> advanced settings – Apply language settings to the welcome screen and new user accounts -> Welcome screen and new user accounts settings

Clock, Language and region -> Language -> advanced settings – Region settings

From that forum:

Please make sure your Country or Region, Language and Speech Language settings are all aligned with one of Cortana’s supported locale configurations. If these OS settings are not aligned, Cortana will not be available. For example, Cortana will not be available if your country is set to United States, but the Windows display language is set to English (United Kingdom).

1. Open Settings. Select Time& language, and then Region & Language.
2. Check the Language (set as default) setting for your Windows display language.　 If your desired language is not available, add your desired language:

• • • Click Add Language.
    • Select your desired language from the list.
    • Select the desired locale, which is the language/country combination.
    • Click on the newly selected locale and select Options.
    • Under Download language pack, click Download.
    • Under Speech, click Download.

• After the downloads are complete (this could take several minutes), return to the Time & Language settings
• Click on your new language and select Set as Default.
• NOTE: If you changed languages, you must sign out of your account and back in for the new setting to take effect.

3. Check the Country or region setting. Make sure the country selected corresponds with the Windows display language set in the Language setting.
4. Return to Settings and Time& language, and then select Speech. Check the Speech language setting, and make sure it’s aligned with the previous settings.

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P.S.
4 Sept 2015 – In the following the Microsoft instructions sent today through the Insider Hub:

Setup Instructions

Cortana is available to Windows Insiders in the following configurations. In order for Cortana to be show up on your system, please ensure you system has the appropriate Country or Region, Language, and Speech Language select for your configuration.

1.  Open Settings. Select Time & language, and then Region & language.

2.  Check the Language (set as default) setting for your Windows display language.  If your desired language is not available, add your desired language:

• Click Add a Language.
• Select your desired language from the list.
• Select the desired locale, which is the language/country combination.
• Click on the newly selected locale and select Options.
• Under Download language pack, click Download.
• Under Speech, click Download.
• After the downloads are complete (this could take several minutes), return to the Time & Language settings.
• Click on your new language and select Set as Default.

NOTE:  IF you changed languages, you must sign out of your account and back in for the new setting to take effect.

3. Check the Country or region setting.  Make sure the country selected corresponds with the Windows display language set in the Language setting.

4. Return to Settings and Time & language, and then select Speech.  Check the Speech language setting, and make sure it’s aligned with the previous settings.

If you are still having issues; you can further verify your settings:

• From Region & Language, under languages is a Related settings section, click on Additional date, time & regional settings, then in the control panel widow that opens:
• Under Language, click Add a language.
• A table of installed languages will display, click Options for the language you are using.
• At the top of the page under Windows display language, click Make default display language (or something like that).
• You will need to sign out for changes to take effect.

Note: If it says Enabled under Windows display language, then this is already done.