When to email recruiter?

[OopsIDidItAgain123]

I’m currently going through the process of moving abroad for work- in a sector with 0 redundancies due to COVID (extremely lucky!) so that isn’t an issue.
I am doing this through a reputable agency who have been involved in such work for 30years, I was offered a position in February for a post abroad but warned there are 30 other ahead of me so it will be a while before I receive my formal offer letter and the visa can be processed. However, I had the interview and was congratulated. So that is in the pipeline.
My recruiter called me 3 weeks ago just to discuss progress on this and to inform me there are similar positions coming up in a neighbouring country soon so would I like my CV to be sent. My CV was sent to this employer in mid March. My recruiter re-assured me she would be in touch as soon as she had any updates/feedback on my application for this country. I have received generalised emails from the agency advertising positions available for this location in the meantime but nothing from my recruiter!
When would it be acceptable to email her politely asking when will I know if I have been successful in being selected for an interview? There are multiple positions available so not fiercely competitive. It’s 2 weeks tomorrow since it was sent. I have only ever worked in my same job in the U.K. so not too experienced on this and working abroad (other than the initial job offer I have for the first location). The recruiter knows I am keen but I don’t want to seem annoying!

Are these two separate recruiters? If you haven't heard from the first for three weeks I think you could ask for an update and when you can expect to hear by now.,

The same recruiter! There’s an overall agency who send generic emails and then my personal recruiter who has a caseload of us and supports once abroad/liases etc. So the recruiter is aware I’m on the waiting list for the first job where I’ve had the interview and that I’m
Waiting to hear if I am successful for obtaining an interview for the second country.
@notdaddycool thank you! I’ll email next Tuesday as that will be 3 weeks since my application was submitted.
It’s very frustrating to receive generic emails advertising such positions but not hear any feedback on my application! The recruiter has otherwise been great and usually responds very quickly

Recruiters work for themselves not you would be my first piece of advice!

Email them by all means but don't expect things to move quickly, particularly with an overseas move. They need the people above them to be available to sign off and move things on.

@idontlikealdi very good point! I hadn’t considered it from that perspective but you’re right.
Is this an ok email to send:

Hi Recruiter,
I was just wondering if you had heard anymore on the X country opportunity such as when we will know if I have been offered an interview and likely interview dates for successful applicants etc? It has only been 3 weeks since my application was submitted but I am very very keen and wasn’t sure what kind of time frame to expect regarding a potential interview if my application is successful. Sorry if you receive many of these emails asking the same question! Thank you again for your time and help,
Best wishes,
Desperate to relocate individual

I don’t want to seem overbearing/annoying!