score:7
Technically an Emergency Travel Document/Emergency Passport IS sufficient to enter the US on the Visa Waiver Program - as long as it is an "e-Passport". From a quick Google search I can't see any mention that UK ETD's are e-passports (although they are "Machine Readable", they do not contain an RFID chip) and presuming that's the case then they can not be used with the US Visa Waiver Program, even if you are just transiting the US.
Your best course of action here is going to be to talk to your local embassy/consulate (ie, the UK consulate in the country you are in). You are going to need to contact them anyway to obtain the ETD, and they will be very familiar with the requirements for transiting the USA using such a document.
It's very likely you will need to obtain a visa of some form to enter the US, most likely a C "Transit" visa. Generally the process to obtain such a visa will take at least 2 weeks (including the time to get the relevant information together, schedule and attend an interview, and then actually wait for your passport/EDT to be returned). However, most consulates also have an "expedited" process to allow you to get an appointment sooner. You may or may not qualify for using this process, but it can't hurt to try. For Bogota, the steps are documented here, although before doing that I'd strongly suggest talking to your own countries consulate as they likely have connections within the US consulate that may allow them to speed up the process.
Obviously the other option is to change your flights to avoid the US on the way back to the UK. That's likely to be expensive, but it may be worth contacting your airline to see what's possible.
Upvote:5
You'll need to have a transit visa for the United States. You can complete the DS-160 application form online, but you'll also need to have an interview at a US embassy or consulate. Because this is on short notice, however, you may also just wish to dial up the local US embassy/consulate and let them know the details of your situation to see if they can provide more specific guidance. The current processing time in Bogota for a C Transit visa (not a visitors visa) is five days, so you should be in luck, though, assuming you already have your ETD or you can get it issued very, very quickly.
You should also contact the British embassy/consulate. It's their job to provide assistance to British citizens and nationals travelling overseas, so if the US authorities are less than helpful, they may be able to help.
Under a special agreement with the United States, holders of a UK passport do not need six extra months of validity to enter the US. I don't know how long the Emergency Travel Document is valid for, but that should not inhibit you transiting via the US.