It might be only my opinion, but from what I can see TGA...

It might be only my opinion, but from what I can see TGA approval does not mean FDA approval. Otherwise all the devices on the TGA list would have been on the FDA list. But some data obtained for TGA approval may be used for FDA approval. TGA currently does not approve products outside healthcare professionals settings, whereas FDA has processes to approve that, as discussed below.

When looking at the templates for FDA submission, it looks like if the company wishes to market the product as OTC (over-the-counter), it will need to carry out a number of studies as follows (hence the reference to the "clinical trial - US" in the timeline):
"

7) Human Usability Study

· Testingshould include a minimum of 100 participants for non-prescription (OTC) testsand 30 participants for prescription only tests, and take place in an actualuse environment or simulated environment. For OTC tests for use at non-laboratory sites, we recommend you splitthe usability study into two sections: 50 participants testing themselves and50 participants testing another person (child or adult, depending on yourintended use population). Forprescription only tests for use on children, you should have 15 of the 30usability participants be parents or legal guardians performing the test ontheir children.

[...]
Weencourage sponsors to submit their usability study protocols and questions forparticipants for FDA review prior to conducting the study. It may be possible to combine the HumanUsability with the Clinical Evaluation; however, this study design does involvemore risk as problems with the instructions for use could lead to a failedclinical study. FDA strongly recommendsyou discuss this option with FDA before design and execution."

8) ClinicalEvaluation:

FDArecommends using natural clinical specimens for the clinical evaluation. Youshould conduct a clinical study to evaluate your device’s performance insymptomatic and asymptomatic individuals. This study design evaluates performance in asymptomatic individuals aswell as symptomatic individuals. Sincethere is no mechanism to limit OTC testing to symptomatic individuals, FDArecommends this study design for all developers requesting an OTC claim. This study design is also recommended fordevelopers requesting prescription use, unless the test is intended to belimited to symptomatic individuals.

1. Testing Sites

The sponsor should attempt to set up a minimum of 2 testing sites toencourage diverse enrollment or recruit for an at home clinical study throughthe internet. [...]

So it looks like the company will first need to talk with the FDA about the design of the study protocols before carrying out the study with the requirements, among others, as per above. Looking at how quick Ellume received approval after its trials and submission, I reckon that if all is good, the submission and approval would happen in October/November. As AnteoTech said, in time for the winter in the U.S.

You can find details and templates for the submission here.
https://www.fda.gov/medical-devices/coronavirus-disease-2019-covid-19-emergency-use-authorizations-medical-devices/in-vitro-diagnostics-euas