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My mother is applying for EUSS following the Lounes route, as a dependent of me (naturalised UK citizen).
Mother: EU and Third Country (TC) citizen, with EU Family visa valid until April 2022.
Me: EU citizen, naturalised UK citizen, TC citizen
TC citizens do no need a visa to enter the UK for up to six months.

In December we sent her passport, my EU and UK passports and other documents to the HO. We have not heard back from them yet and my mother has not yet received an invitation to provide biometrics.
My mother needs to travel to the EU in the next few weeks and I was planning on asking for the documents to be returned. My question is:
Is she able to travel while waiting for the outcome of her application?
If her passport is not returned on time, how could she leave the UK?
Her TC passport has no entry stamp and she has an EU ID, but we know that EU IDs are no longer acceptable for travel etc etc.

Also, will she be able to return without her EU passport?
She could enter the country with her TC passport, but will that be a problem later on?

Any comments on this would be helpful.

I cannot comment on your other questions, but regarding the below:

LillyPearl wrote: ↑

Sun Jan 02, 2022 3:20 pm

Is she able to travel while waiting for the outcome of her application?

If you are asking if her application is going to be cancelled just because she travelled abroad, then the answer is no. Unlike every other application under UK immigration rules, travel abroad does NOT in itself cancel applications for the EU Settlement Scheme.

As per caseworker guidance, page 68:

Travel outside the Common Travel Area
An application made under Appendix EU will not be treated as automatically withdrawn if the applicant travels outside the Common Travel Area before the application has been decided.

Please note that this does not mean that travelling abroad won't have its potential issues as you correctly suggest in your other questions.