Answers, not necessarily in order.
1. There are a number of significant tax benefits for dependent children that you only qualify for if you file an on-time return with SSNs for your children. If you get the SSNs later and file an amended return, these benefits won't be paid. For this year, an on-time return means filed before July 15, 2020; OR filed before October 15 but only if you have a valid extension. To get a valid extension until October 15, you must file form 4868. Usually, this form can't be e-filed after the regular deadline of April 15. I don't know if this has been extended. Overseas taxpayers get an automatic 2 month extension to June 15, and you can file form 4868 by mail after April 15 and before June 15 by using the overseas extension method (which I think, is to write your request on the top of the form, but check the instructions.) This year, it might be possible to still e-file a valid form 4686 even though it is after April 15 because the mail filing deadline was extended. Either way, if you plan to file a tax return with your dependents, you need a valid extension to October 15 and to actually get the SSNs before you file an on time return. https://www.irs.gov/forms-pubs/extension-of-time-to-file-your-tax-return.
For 2020, an on-time return will probably be one that is filed by April 15, 2021, or by October 15, 2021 as long as you file for the extension before April 15.
2. If your children are eligible for SSNs, don't bother applying for ITINs. You won't get any extra credits or benefits, and the application will probably be denied anyway. Just apply for the SSN. The US Consulate wherever you are shouldn't be completely shut down, there should be some kind of procedure since I would hope this to be considered "essential".
3. The payment is actually an advance payment of a new credit that will be added to the 2020 tax return. The IRS determines the amount of the advance using 2018 or 2019 income. When you file a 2020 return, the payment is recalculated, and if you qualify for more than you received in April or May, the extra is added to your refund. (If you qualify for less than you received, you don't have to pay it back.)
I don't know how this will work if you are still a non-filer for 2020 as well. And the situation is still very fluid. But it is at least possible that you can file something for 2020 in the spring of 2021 and get the payment, if your children have SSNs by then.
4. Are you really a non-filer? As a US citizen, you are required to pay for that privilege by filing a US tax return and paying income tax on your world-wide income, even if you live outside the US. (You can claim a credit or deduction if you also pay foreign income tax on the same income.) In some cases you can use the foreign earned income exclusion to not pay US tax on your income earned from work, but you still have to pay US tax on unearned income (passive income, investments, bank interest, capital gains, dividends etc.). So be careful that you really are a non-filer. If you have income that is subject to US tax and you aren't paying it, you may not want to call attention to yourself. You may want to have your situation reviewed by an accountant.
5. If you actually need to file a tax return for 2019 after all, you should wait until you get the SSNs, even if you are late (past even the October 15 deadline). You will not get all the benefits of your children having SSNs but you will get some benefit for listing them as dependents and, as I said, you shouldn't try to file faster by applying for ITINs.