Will I be eligible for an Emergency Travel Document?

You do not need to renew your passport or obtain an emergency travel document to travel from the US to the UK as a British Citizen.

You will be admitted to the United Kingdom with your expired passport.

Information as of  22JUN15 / 2124 UTC
National United Kingdom (GB)    /Residence USA (US)
Embarkation USA (US)
Destination United Kingdom (GB)
United Kingdom (GB)

Passport required.
- Expired British passport (no time limit) endorsed  British
Citizen  is accepted.

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Your difficulty is returning to the USA. However, it will be a lot quicker and cheaper to take advantage of the renewal process while in the United Kingdom. See https://www.gov.uk/get-a-passport-urgently

You can see what appointment slots are available in advance and book a convenient slot before you leave the USA (potentially before you book your travel). You can use the premium one day service to obtain your replacement passport on the same day as your interview. (That assumes your replacement gets authorised quickly; and although there is no reason to expect a simple renewal to take long, you may wish to have a few days of padding. Personally I would want to start this process on the first day of arrival.)

Alternatively---if the renewal is taking too long---you may apply for an Emergency Travel Document while in the UK to travel to the US. If you are a permanent US resident, it should be accepted for entry, although you may wish to confirm this with the US Department of Homeland Security.

Scroll down to "passport exemptions":

Information as of  22JUN15 / 2127 UTC
National United Kingdom (GB)    /Residence USA (US)
Embarkation United Kingdom (GB) /Destination USA (US)
USA (US)

Passport required.
- Passports and/or passport replacing documents issued to
residents of the USA must be valid on arrival.
- British passports endorsed  British Citizen  must be valid
for the period of intended stay.
- British passports endorsed  British Overseas Territories
Citizen  issued by Bermuda must be valid for the period of
intended stay.
Passport Exemptions:
- Residents of the USA when holding:
- Permanent Resident/Resident Alien Card (Form I-551);
- US "Permit to Re-Enter" (Form I-327);
- Holders of a Form I-512 ( Authorization for Parole of an
Alien into the United States").
- Holders of temporary or emergency passports.
- Holders of a Transportation Letter:

Visa required, except for Residents of the USA if holding:
- Permanent Resident/Resident Alien Card (Form I-551);
- US "Permit to Re-Enter" (Form I-327);
- Employment Authorization Card (Form I-776), provided
endorsed "Serves as I-512 Advance Parole" or "Valid for
Reentry to US";
Visa required, except for Holders of a Form I-512
( Authorization for Parole of an Alien into the United
States").
Visa required, except for Holders of a  Transportation
Letter":
Minors:
- Children, up to/incl. 15 years of age, being nationals of
United Kingdom, are not allowed to enter the USA on their
parents passports.
- Minors traveling unaccompanied, or accompanied by one parent
or a person other than parent/legal guardian,
.
- Visa required, except for minors seeking residency; provided


Additional Information:
- An expired Permanent Resident/Resident Alien Card (Form
I-551) with a 10-years validity at time of issuance, is

- If the passport shows a stamp "processed for I-551 temporary
evidence of lawful admittance for permanent residence valid
until receipt of I-551 employment authorized , such stamp is
to be considered Form I-551.
- Transportation letters issued to lawful permanent residents
of the US are being replaced by "Lincoln Boarding Foils".

Warning:
- Passengers who have visited Guinea, Liberia or Sierra Leone
within the last 21 days must enter the US at Atlanta (ATL),
Chicago (ORD), Newark (EWR), New York (JFK) or Washington
(IAD)  and are subject to screening and medical checks upon
arrival.

USA (US)

Vaccinations not required.

Warning:
- Those entering on a K-1 or K-2 visa must hold a medical form
FS398 and X-ray plates of chest (full size not necessary).

This information is from TIMATIC, the computer system that the airlines use to decide whether they can let you on the plane. There is free access to it here: http://www.staralliance.com/en/services/visa-and-health/