A general guide to getting a UK passport can be found here.
Passport applications are dealt with by Her Majesty's Passport Office (ONL - OVS), Millburngate House, Durham, DH97 1PA, United Kingdom. They are no longer dealt with by the hub in Washington DC.
Best advice is to not book travel until you have the passport in your hand.
In the US, go to here and in Canada, go here and click on the passports tab.
- US sized photo's are ok, they'll get resized digitally by the UK Passport Service. Pay attention to facial expressions though - no smiling please!
- Counter signature, this only needs to be done if you significantly look different from your previous passport or it is your first. A British person does not have to do the signature, just someone of standing such as a doctor, teacher, lawyer etc.
- Child passport applications normally need countersignature.
- Normally countersignature needs to have known you for 2 years, however, if you have lived in the USA/Canada for less than that time, the UK Passport Service in Washington will usually be flexible.
- If your existing passport is expired less than 10 years and you are including it in the application then you normally do not need to provide documentation proving your British citizenship (unless the passport is an old-style blue one).
- If you are British by Home Office registration or naturalisation, they do want to see a photocopy of your citizenship certificate, normally.
- If your last passport was a child passport and you're now applying for an adult passport, documentation is normally needed.
- The passport can be renewed at any time, but only any valid time remaining within 9 months or expiration will be transferred to the new passport.
US Birth Certificates
- Many, if not all, US states allow an adopted person to apply for a new birth certificate showing adoptive parents as natural parents (this does not apply in Canada).
- An adoptive parent link does not confer automatic British citizenship (although child can be registered as British until age 18)
- Hence, the UK Passport Service in Washington does not accept US birth certificates as proof of a link to a natural parent, unless the birth certificate was issued within 3 months of birth.
- If you have replaced your birth certificate, you will need additional evidence to show you were not adopted. This can include hospital records from when you were born. Alternatively, you may need to provide additional documentation from the state birth registry confirming that your birth certificate has never been reissued pursuant to adoption.
- You do not need to get the actual certificate of registration to save some money. You can get that after the following September directly from the GRO for £9.25 here instead of the $107 per copy, plus $15 shipping that the cost would be direct from the Embassy here.
You don't normally need to send in a foreign (US or other) passport, however they do like to see a photocopy of a US/other passport, if possible, as proof of identity.
Emergency Travel Documents
If your passport is lost or stolen and you need a new one quickly: 
Right of Abode Stamp
If you have a complex claim to British nationality and you have never before held a British passport, it might be quicker to apply for a Right of Abode stamp in your US/Canadian passport instead. This lets you live in Britain and you can get a British passport in the UK later on. See UK Visas
If you were applying for a first passport for your baby born in the US, you could also send in the application for a consular birth certificate at the same time if you wished. Also, as a US citizen, the baby would need a US passport to travel and for the purposes of a holiday to the UK, they could just use their US passport. If you wished to get them a UK passport, you could then do that while on holiday to the UK, Fast Track does take a week and currently costs £87 here while getting them a first time passport in the US currently costs $167 including the $33 courier fee here and can take up to 6 weeks.
You can include up to 4 passport requests/renewals in the same packet to save on return postage - BUT check each form very carefully as they will return ALL the packet even if only one set is wrong.
To replace a UK birth certificate, contact GRO England & Wales, GRO Scotland, or GRO Northern Ireland. To replace Home Office naturalisation or registration certificates, contact the Home Office or, if the certificate was issued before 1986, the National Archives.
If your previous British passport was for a status other than British citizen (eg British Overseas citizen) and you have since become a British citizen, you need to make a "first time" application to get a British citizen passport.
An expired British citizen passport is acceptable to enter the United Kingdom, if you are flying directly, although you should never rely on an airline accepting this and it should only be considered in cases of real emergency: Travel planner - Delta Airlines
If you were born before September 2, 1929, your British passport should be free: Free passports for wartime generation
If you have to apply for British citizenship first - eg through the UKM or MN1 forms to the Home Office - you have to wait until your citizenship is granted before you can apply for a British passport. You cannot apply at the same time.
Where to send the application
Send it to:
Her Majesty's Passport Office (ONL - OVS), Millburngate House, Durham, DH97 1PA, United Kingdom.