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

11. (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;
    ...

(Emphasis added.)

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."

In principle you should be able to use your expired passport as evidence that you're a citizen of Portugal and therefore enjoy free movement within the EU. Directive 2004/38/EC states in article 5:

4. Where a Union citizen, or a family member who is not a national of a Member State, does not have the necessary travel documents or, if required, the necessary visas, the Member State concerned shall, before turning them back, give such persons every reasonable opportunity to obtain the necessary documents or have them brought to them within a reasonable period of time or to corroborate or prove by other means that they are covered by the right of free movement and residence.

(my emphasis).

It doesn't say anywhere that they can't completely ruin your day, and/or detain you while, for example, they try to reach the Portuguese authorities to verify whether you have lost citizenship after the passport expired, etc. But eventually you ought to be able to get in.

An additional piece of paper saying that you have applied for a new passport that you don't actually have yet will probably not help much, or at all, relative to a recently expired one.

If you have a reasonable expectation that your new passport will arrive at your UK address while you're away, it might be a good idea to arrange for a trusted friend or family member to be ready to bring it to the airport if worst comes to worst.