Yes she may travel provided that she is carrying both passports with her. I would suggest that she staples the two passports together.
She can also apply for a TOC (Transfer of Conditions) however this would require her to submit all of her documents used to obtain the ILR in the first place and submit a fee.
I hope this answers your question. If so, kindly rate the answer and provide feedback.
If you have any further questions, please do not hesitate to ask.
You can have a immigration lawyer provide you with a written opinion. Save for that, all of the information is available on the UK Home Office website.
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