Travel with Expired Greek Passport from UK to Greece and Return

See the answer by @Chris Melville, he is in the business and knows what he talks about.

You still have two months before Christmas, best to initiate the passport or ID card replacement right now, with a request for fast service as the earlier try was not handled as it should be.

If you can not get your passport or ID card replaced in time while not in Greece, you may want to make sure your stay in Greece will be long enough to handle it.
Leaving the UK is not a problem as long as your paperwork allows you to enter Greece and the airline can look up in the database that your actual paperwork is acceptable.

Most countries have a fast (one day often) service for emergencies and for people who need and are willing to pay for their passports in a short time.
As I do not read the language I have not done a search for that service, you can look online.

If neither option are working for you, you may have to forgo your travel to Greece or lengthen your stay in Greece so you can get your passport sorted before returning to the UK.
The UK has no reason to let you in if you can not show the right paperwork and airlines know that and will not transport you to the UK unless you have shown an acceptable ID, which in your case is only a passport or a Greece ID card.

While you blame (likely rightly) the passport office, getting them to pay for the mistake is completely separate from traveling. And it is very likely they will not pay for any of the cost you have to make.

"My Greek ID is in a bad condition and will not be accepted"

How bad? Do not assume that it won't be accepted. Even if it's a bit tatty, as long as your photograph still identifies you - and contains no signs of tampering or fraud - then it will be accepted.

Possibly the MRZ (Machine-Readable Zone) won't work if the card is too spoiled. But this isn't a problem. An Immigration Officer is obliged to examine the document when you produce it to them.

I used to work as a UK Immigration Officer myself, and saw thousands of poor-quality ID cards from many countries. Cards which had been stored in someone's back pocket, dirty and dog-eared. Even cards which had been through the wash! However as long as the photograph still identifies you, and the security features are present, then the card may still be used. If you're really concerned about the condition of your ID card, you can also present the card in conjunction with your expired passport. Therefore even if the passport itself has expired, it will still establish your Identity and Nationality. These are the two criteria against which admission is measured by the immigration authorities.

If I were you, I would change my flight. How much? Ask the authorities how long they expect it to take.

My Hellenic mother had a similar experience with an employee touching her photograph, and guess what, nothing practical she could do about it. I don't know if there is an elaborate and hairy process of chasing this, but it's probably not worth the effort, time and money, if any. Explain to your employer the situation.

You learnt it the hard way, but please make sure to renew your passport in time next time, typically 6-9 months before its expiry date.

It's not certain that if the OP starts the procedure now, he will have a passport before his travel to Greece. The Greek embassies in Europe urge Greeks to go back home in order to renew their passport due to the long waiting times (reference: personal experience). And a fast procedure only exists for serious reasons like health problems.

The only loophole I can find, is that you replace your old ID card once you are in Greece - you will receive it on the same day - and then travel back using this as a backup. While you are in Greece you can also request a new passport which in case it's not ready when you are leaving, a family member or a friend of yours will receive on your behalf and can mail it to you. So you can have a passport for sure in the beginning of the new year.

I would like to respectfully suggest that you change your approach.

You write:

the authorities must give me every chance to get one within a reasonable time limit or produce other satisfactory evidence of your identity and citizenship...

Later, you say:

...it is not my fault and I should deserve a compensation as I am gonna miss the flight and have to pay extra 90 pounds for the renewal (new process).

The first misstates your experience, and the second is incorrect.

First, airlines are required to return to the departure point at the airline's expense (although the airline may try to collect the fare from the passenger) every passenger who is refused entry at a destination or transit point. Thus, airlines are extremely careful to check every passenger's documentation; airlines use the Timatic database for this purpose. Because there are an almost infinite number of combinations of factors (which countries, what kind of entry or transit, passenger citizenship, passenger residency, visas held, age, etc.), reference to a standard database makes sense.

If the airline thinks you won't be admitted at your transit point or destination, you will not be allowed to board the flight. You can see your own document requirements by entering your data into a Timatic portal; here's one, there are others as well. Put your data into Timatic: you'll see that with your current documents you can fly from the UK to Greece, but you'll be barred from flying back to the UK.

The airline will rely its Terms & Conditions of Carriage, to which a ticket holder (whether a purchaser or only a user) is deemed to agree by presenting themselves as a passenger. Without exception, these T&Cs provide that it is the passenger's sole responsibility to have the correct travel documents to make the flight. This is not the airline's job, it is yours.

Second, the government (whether of Greece or the UK) has no duty to keep track of what documents you hold, what kind of condition those documents are in, or when they expire. This is, again, your job alone.

Here, you didn't pay attention and left passport renewal and Greece national ID replacement until very late in the game. While the time required was increased because of a error by the Greek bureaucracy, in the real world...things happen. It would be delightful if the Greek bureaucracy responded by accelerating your Passport renewal, but bureaucracies are not known for timely and moral responses. A more prudent traveler would have presented the application earlier, in case an unexpected issue arose.

You can complain and whine about how poorly you've been treated, but it won't resolve your issue of insufficient documentation to re-enter the UK, and it will certainly not help you grow into a more capable traveler.

An effective reaction, on the other hand, would be to follow the good advice given by @Willeke and @gsamaras and @Chris Melville in their Answers. Reframing your experience as "I screwed up this time, but I'll pay more attention in the future so this won't happen again" would be a much more productive strategy.

When my passport expired it took them 2 weeks to renew it for me (I'm from an EU country and renewed it in UK). Even then, you have like 2 months before. You can also pay a bit more and get it done faster. Not much more you can do.