I don't think it matters. This page:
http://www.ukba.homeoffice.gov.uk/briti ... documents/
- lists various documents which might be submitted as alternative to a passport as evidence of nationality and identity, including, for example a birth certificate.
Further on it suggests that if you can't produce a passport including the ILR stamp, you could submit the HO letter conferring ILR.
All this suggests to me that a recently-expired passport would be sufficient to support an application.
And most certainly, your ILR doesn't lapse simply because your passport expires.