You didn't say who has the original Will. Typically the court will want the original Will, although if the uncooperative Co-Executor has the original and someone else has a copy, you can file the copy and ask the court to order production of the original.
The first thing to understand is that the Will only nominates the Executors, it does not appoint them. If one of the Executors is unwilling to start a probate, anyone else can start probate proceedings. It would be logical for the Co-Executor to be the one to initiate the probate if that person is willing to handle things. In the application, simply explain that the Co-Executor has been resistant. The court will then make a decision about who to appoint.
In most states there is no deadline by which a probate needs to be started, but an uncooperative Co-Executor does not need to stand in the way. The sooner it starts, the sooner you can put everything behind you!
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
This answer is not intended to provide you with specific legal advice regarding your situation, or to create any attorney-client relationship.
There is not a specific time limit on offering a Will for probate in Georgia, as long as no one has started such a proceeding using another Will. If one of two Co-Executors named in a Will does not want to serve, then the other Co-Executor should still be able to file a Petition to have the Will admitted to probate. The first respondent was correct in stating that the court will normally want the original Will. To offer the Will for probate, the nominated Co-Executor who does want to serve will need to prepare either a Petition to Probate Will in Common Form or a Petition to Probate Will in Solemn Form. Either petition basically names the decedent's heirs and states that the Will is the decedent's Will. A Solemn Form petition also has places where the heirs are asked to consent to having the Will admitted to probate. If one or more of the heirs does not sign the Petition consenting to have the Will admitted to probate, then the non-consenting heirs generally need to receive notice and they then have a short period of time in which to make any objections, or the Petition will be granted. A Common Form Petition does not have the consent of the heirs, but they have a much longer period of time to object to the probate under a Common Form Petition, and it is generally best to file for Solemn Form unless for some reason it will be difficult to get all heirs to consent and it is very important to have someone appointed as Executor quickly (such as where the decedent owned a business and there is no one who can continue to operate the business until an Executor is appointed). The Petition would also need to state the reason why the Will nominates Co-Executors but only one of them is seeking appointment as the Executor. While in Georgia someone can offer a Will for probate without an attorney, it is generally helpful to have the help of an experienced probate attorney.