OKAY. So my first thought was to consider what protocol the U.S. has for a Yellowstone Eruption since the supervolcano is in the same region and an eruption would require millions of people to evacuate.
This source: https://pubs.usgs.gov/circ/1351/downloads/circ1351_v2.pdf has all of their PRE planning explained as far as I can tell. Who would be watching, what alerts they give, and such. They have a large team of scientists consistently watching data.
Now, with this theoretical <PERSON> attack, that is a direct threat to American citizens right over the border. This leads me to believe it would be a combination of military personnel and the Department of Homeland Security who would work on slowing down Godzilla and attempting to take it out. https://www.dhs.gov/about-dhs#:~:text=The%20Department%20of%20Homeland%20Security,analyst%20to%20chemical%20facility%20inspector.
This guide to Mass Evacuation by the National Governor's Association highlights some important things.
Firstly, non-governmental organizations would have to be promptly contacted and evacuated with the main population. These include organizations like the Red Cross and the National Voluntary Organizations Active in disaster. These group would help provide evacuees with necessary medial/first-aid treatment and other necessities that such a disaster may create need for.
Secondly, previous to this event, shelters likely will have been created and evacuation plans at least known by those helping aide evacuation. When <PERSON> first appears on the American radar, an evacuation order would likely be sent out. The farther people are from the coast, the more likely they are to be more prepared since they will have more time to do so.
The above link says: Among the basic elements evacuation plans include are the following:  - Hazards and vulnerabilities that can cause evacuations;
 - Decision-making authority;
 - Types of evacuations (mandatory or voluntary);
 - Clear evacuation terminology coupled with legal implications;
 - Interagency and multijurisdictional coordination and communication;
 - Specific state agency roles and responsibilities;
 - Concepts of operations, including timing of triggers for evacuation;
 - Mass care and sheltering capability and implementation;
 - Reverse-lane procedures (if necessary);
 - Logistical contract support needs (public transit, buses, ambulances); and
 - Beneficial mutual aid support agreements with counties, neighboring states, and NGOs.
Other Evacuation things: 1. Phasing (page 8): helps get things done orderly. 827
This would likely be done in a situation with more forecasted warning as this takes time. 2. Contra-Flow Plans (page 9): allow all roads to essentially be one direction to help speed up the process of outward flow from a place. 3. Shelter-In-Place (page 9): when staying home is safer and likely to keep people under better protection than attempting to flee. 4. Emergency Powers (page 11): who's in charge during an evacuation/crisis
Pages 13-14 have a list of important legal figures from state to state.
Alerting systems would include: television, radio, road signs, texts, and social media, systems like the Integrated Public Alert and Warning System to push information out to the public by using them to issue emergency alerts, including wireless emergency alerts, the National Oceanic and Atmospheric Administration (NOAA) Weather Radio, and other public alerting systems
ON EVACUATION TERMS
Terms such as “mandatory,” “voluntary,” “partial,” and “recommended” have specific legal implications, and the precise meaning of those terms varies from state to state. For example, some states have arrest authority for citizens who refuse to heed a mandatory evacuation. In some states, public safety officials have a reduced liability to respond to 911 calls during mandatory evacuations. Governors are encouraged to review terminology and work with FEMA and state, local, and tribal emergency management agencies to create effective and clear terminology consistent with the National Incident Management System. The terminology should be phrased in a manner that causes citizens in imminent danger to react quickly.
As defined in the Comprehensive Preparedness Guide 101 (CPG 101):7
- A voluntary evacuation is a warning to persons within a designated area that a threat to life and property exists or is likely to exist in the immediate future. Individuals issued this type of warning or order are not required to evacuate; however, it would be to their advantage to do so.
 - A mandatory or directed evacuation is a warning to persons within the designated area that an imminent threat to life and property exists, and individuals must evacuate in accordance with the instructions of local officials. 827