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If you do not have a suitable travel document but need to leave the UK (either by choice, or for a 'voluntary departure' or because the Home Office is removing you), they will not rely on someone else to provide a document. They will issue you an IS137 One Way Travel Document. This is essentially a piece of paper with a passport photo stapled to it and some biographic details.
The Home Office state that it is the holder's responsibility to ensure they are able to enter their destination country. I believe in cases of removal, the Home Office will have an agreement in place that you will be accepted.