I keep getting messages saying that my stolen phone has been activated with a new SIM card

First off, if you have access to your iCloud account still, change the password if you haven't already. Try using Find my iPhone on someone else's phone (Like a family member, etc) to track down where the thieves are. If you have Find My iPhone turned off, then that doesn't leave much room.

Try to call Apple Support again. They may be backed up with a lot of Support Calls.

I hope you get your phone back and your problems situated.

Best of luck,

Mac

See these security-related documents, especially the first two for now.

If you think your Apple ID has been compromised https://support.apple.com/en-us/HT204145. From the link, among other things, Change your Apple ID password and choose a strong password.

Security and your Apple ID https://support.apple.com/en-us/HT201303

iCloud security overview  https://support.apple.com/en-us/HT202303

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And, if you can, Check your Apple ID device list (https://support.apple.com/en-us/HT205064) and remove any device that shouldn't be signed in using the steps in the link.

If there are none that shouldn’t be signed in, then you should not have a problem at this point. But, if an unauthorized device is signed in, Change your Apple ID password https://support.apple.com/en-us/HT201355 

Here is What to do after you change your Apple ID or password https://support.apple.com/en-us/HT204071

For Apple Support help with your problem, use the support links below. No that I know that you currently are trying to contact Apple support.

Get a call from Apple support to help solve your problem  https://getsupport.apple.com/?caller=kbase&PGF=PGF63005&category_id=SC0245&symptom_id=23362

Apple ID Support https://support.apple.com/apple-id

Apple Support Twitter https://twitter.com/AppleSupport

Apple ID account security for country-specific and region-specific

help https://support.apple.com/en-us/HT204169