How to speed up processing time for UK transit visa from USA?

Hello Everyone,

Its my very first time posting here so I hope i am doing this right. I am a Jamaican Citizen holding a green card living in the US. My wife (who is a US citizen) and I are planning a trip to go to Europe to celebrate our wedding anniversary. We booked our tickets and we are set to leave on July 16, 2019. We are excited to go Europe because this will be our first time.
Seeing that my wife is a US Citizen, she wont need anything but her passport but for me it is required that I get two visas, a Schengen visa because we will be spending some time schengen area and a UK transit visa, seeing that we are going to change planes in Heathrow to go to the schengen area. I already got my schengen visa but its the UK transit visa that got me a little worried. On June 7, 2019 my passport was delivered with all the supporting documentations and flight itinerary. Only got a note that it was received and that was it so I emailed them to find out if there was any problem with my visa application because I heard it should take 15 working days to get the visa in hand. This was the response I got:

 "Thank you for contacting UK Visas and Immigration International Contact Centre.
With regards to your enquiry about the status of your application, I can confirm that we have made a decision on your application. We cannot discuss the outcome of your application on-line or by telephone. However the visa application centre will be in contact with you in regards to collection or delivery of your passport.
We strongly recommend that you do not buy a non-refundable, non-flexible ticket to the UK, until your visa application has been approved and you have received your documents."

We are very concerned as its been almost a month since they've received my application and I cant track the progress anywhere. Can anyone give some advise on this matter? 
Has anyone every missed their flight because the visa came late? Is there a way to speed up the application?

Thank You so much for taking the time to read and reply. It is very much appreciated.

Kind Regards

Exemptions

You don’t need a visa if you have one of the following:

• a valid USA permanent residence card issued by the USA on or after 21 April 1998

Source: https://www.gov.uk/check-uk-visa/y/jamaica/transit/somewhere_else/no

Enjoy your trip! 

well thats a relief! The sad part is the UK embassy in New York have my passport. wish I had checked that before i sent off my passport!

well thats a relief! The sad part is the UK embassy in New York have my passport. wish I had checked that before i sent off my passport!

You can request to cancel your application and get your passport and documents back, but I'm not sure they will be returned in time for you to travel, as it can take 10 working days to receive them (and it's the weekend now):
https://www.gov.uk/visa-documents-returned

So basically am at their mercy right now. I cant speed it up and if I cancel i might miss my flight
Thank You so much for your prompt replies. I really appreciate it.

Just let you all know. I've received my passport back 4 days prior to my trip! Thank you all for your replies!

Just let you all know. I've received my passport back 4 days prior to my trip! Thank you all for your replies!

That's great news . I'm glad it all worked out in the end.

I hope your trip went well .