Customer initially calls the bank to report an error. Bank gives the customer the phone number to call the Third Party if the complaint is over the threshold for submitting chargebacks.
When does a Notice of Error clock start if a bank is using a Third-Party to resolve their complaints?

In my opinion, notice is given when the customer contacts the bank. Whether or not a third party is used is irrelevant.

I agree. Reg E makes it clear that all the customer has to do is notify the bank. Telling them to call someone else is burdening the customer (in my opinion). If they don't, you've still been notified and the clock is ticking.

§1005.6(b)(5):
Notice to a financial institution is given when a consumer takes steps reasonably necessary to provide the institution with the pertinent information, whether or not a particular employee or agent of the institution actually receives the information.
The consumer may notify the institution in person, by telephone, or in writing.

§1005.11(b)(1):
A financial institution shall comply with the requirements of this section with respect to any oral or written notice of error from the consumer that . . .Indicates why the consumer believes an error exists and includes to the extent possible the type, date, and amount of the error, except for requests described in paragraph (a)(1)(vii) of this section.

I agree with David that telling a customer who is trying to dispute an unauthorized transaction is burdening the customer. The concern with this is not only a violation of Regulation E, but that this could be considered a UDAAP violation and result in fairly substantial UDAAP fines. The FDIC hit several banks a few years back who were requiring a police report before they would conduct an investigation. To me, requiring a customer to contact a merchant before the bank will conduct an investigation is the same problem - it is requiring the customer to jump through hoops that the regulation does not require and hoops that go against what the bank disclosed in their Reg E disclosure.

While I haven't seen any UDAAP fines for Reg E disputes from any regulator except the FDIC, the CFPB noted the following as a common violation in their 2014 Fall edition of Supervisory Highlights:

"At one or more financial institutions, examiners found that customers who complained about unauthorized transactions were told they must first contact the merchant, where applicable, before an investigation would begin."

They conclude this common violation by stating the following:
"A financial institution cannot deny an error claim on the basis of a consumer failing to provide additional information, or require the consumer to contact the merchant involved first."

The common violation found in the CFPB Supervisory Highlights can be found in section 2.3.1 here: https://files.consumerfinance.gov/f/201410_cfpb_supervisory-highlights_fall-2014.pdf