Adding to Henning Makholm's answer, which gives the EU directive, that directive is not necessarily helpful if the UK interprets it poorly. Here's what the UK has to say on the matter, according to the guidance on EEA nationals:
EUN1.13 Alternative evidence of nationality and identity
Regulation 29A allows ECOs to accept alternative evidence of identity where the applicant is unable to provide a valid ID card issued by an EEA member state or valid passport due to circumstances beyond their control.
Each case must be considered on its individual merits and reference must be made to an Entry Clearance Manager in all instances.
Where a person claims to be unable to provide the specified documents relating to cost or inconvenience, it would not be appropriate to accept alternative evidence.
Regulation 29A does not apply to applications for admission under regulation 11, as regulation 11 already contains a provision allowing an applicant to establish his or her right to enter by other means.
So what do the regulations say? From http://www.eearegulations.co.uk/Latest:
Right of admission to the United Kingdom
- (1) An EEA national must be admitted to the United Kingdom if he produces on arrival a valid national identity card or passport issued by an EEA State.
(4) Before an immigration officer refuses admission to the United Kingdom to a person under this regulation because the person does not produce on arrival a document mentioned in paragraph (1) or (2), the immigration officer must give the person every reasonable opportunity to obtain the document or have it brought to him within a reasonable period of time or to prove by other means that he is—
(a) an EEA national;
Alternative evidence of identity and nationality
29A. (1) Subject to paragraph (2), where a provision of these Regulations requires a person to hold or produce a valid identity card issued by an EEA State or a valid passport the Secretary of State may accept alternative evidence of identity and nationality where the person is unable to obtain or produce the required document due to circumstances beyond his or her control.
(2) This regulation does not apply to regulation 11
I conclude that you're likely to be okay, but you might run afoul of the paragraph stating that it's not appropriate to accept alternative evidence when asked to do so for reasons of "cost or inconvenience."