Fortunately, you do not need to assess blame or fault. Nor do UA or CM. It is solely the responsibility of the last delivering carrier to get your bags to you and to pay for interim expenses and ultimately for loss or damage. That is CM and CM alone.
Thus, it does not matter what did or did not happen at DEN (nor frankly is it remotely possible for an agent at UIO or a phone agent to know who left what where and who did or did not deliver what). In short, both carriers are simply trying to get rid of you. UA should simply tell you to deal with CM and CM should deal with you and be done with it.
Stop calling UA, it has nothing to do with this. File a report with CM, make sure that CM has your delivery address & a valid mobile. If you are going to be incommunicado for a week, make arrangements with a friend back in the US to act for you. Provide their name & number and provide the friend with instructions, e.g., whether the bags should be delivered to a hotel to which you are returning or back to the US.
In the meantime, as this is an international flight, it is covered by the Montreal Convention. Each of you should purchase the necessities and keep receipts. Necessities will be more here because you are gone for 7 days. The maximum under the Convention is approximately $1,650 per passenger. If the bags are not recovered, that will be the maximum as well and any interim amounts will be payable as well.
Insurance may help here, but as it appears that if the bags are only delayed by a day and you are gone for 7 more, you would likely only collect the one day. Thus, this may be one of the few times where you do better through the carrier.