"Barts and The London NHS Trust" - Is there a standard way to abbreviate it?
There is a standard way, especially in legal documents but possible everywhere.
Example
Barts and The London NHS Trust is a great organisation, I think so anyway, blah blah blag blah. Anyway, as I was saying, the Trust does a lot of good in the world. The Trust is responsible for …
In a legal document, it is specifically noted that such an abbreviation will be used. In an ordinary, perhaps, journalistic, article, it is assumed that the reader will know that the Trust, refers to the one that is the main subject.
Notes
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I used bold to emphasise "the Trust". That is not needed in an actual document.
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Use upper case for Trust, as it is part of the name.
To avoid repetition of complex or long items, it is common to use abbreviations or referencing pronouns, etc. Examination of existing texts related to the topic can be instructive.
The Lord High Executioner, Sir Charles Chopper, gave a speech on Thursday in which he said that he was opposed to capital punishment. He said that decapitation was no deterrent to the type of criminal prepared to kill. Furthermore, the LHE said, the expense and trouble of purchasing, sharpening, and storing axes was a burden on the finances of the state. Sir Charles said that he would be happy to see his role become 'largely ceremonial'.
This practice is sometimes called 'elegant variation', and can be overdone. Modern style guides either suggest avoiding it, or keeping it to a minimum. A famous example is often quoted where 'carrot' was replaced by 'popular orange vegetable'. This sort of thing is a misuse, especially where brevity is being aimed for.
Elegant variation: the good and the bad