Why wasn't Angelica Schuyler in danger in London despite supporting revolution?

While Angelica was presumably a supporter of the revolution, she certainly wasn't one of its principals. Trying to blame a woman who wasn't plausibly its cause for the loss would have been extremely ungentlemanly by the standards of the time. It would also have drawn attention to the loss of the Revolutionary War, which would have made Britain look weak at a time when it was actually doing reasonably well.

Overall, seeking revenge on someone involved in the American Revolution almost a decade later would have made George III look petty and weak. This is a really bad idea for a constitutional monarch, and I doubt he ever considered it.

In Great Britain there was already a tradition of the loyal opposition, politicians not in power who hoped to win elections and replace the party in power with their own party entirely peacefully and legally.

So being opposed to the policies of the party currently in power was not exactly considered to be against the law.

William Pitt (1708-1778) The great prime minister who led Britain to victory in the French and Indian War or Seven Years War, opposed the policies of the British government in the American colonies and the war against them in the last years of his life.

It may be noted that all of the Hanoverian kings had bad relations with their fathers before them and with their sons and heirs after them. Every son and heir of a Hanoverian king formed an opposition group hoping to gain power when the old king died and the heir inherited the throne.

The only heir who was an exception was King George III, whose father Frederick Lewis, Prince of Wales died before his father King George II, and when the future George III was just a boy. King George III grew up to be devoted to his family and had a mostly happy family life.

George III's son George, Prince of Wales, the future George IV, was no exception to the rule, and supported the Whig party in opposition to the Tory party supported by his father. George III may have been angry at his son's opposition - including possibly opposition to the war in America - but he wasn't going to have his son and heir arrested for treason.

The Prince of Wales raised his daughter Princes Charlotte (1796-1817) as a Whig but that didn't affect the love between the king and his granddaughter.

After what was considered the decisive defeat at Yorktown in 1781 the independence of the USA was recognized by Britain in 1783. And in 1785 the first US ambassador to Britain was none other than John Adams himself.

It is said that when Adams was present to the King, George III said that he had been the last to consent to the separation of the colonies but was the first to wish for a friendly relationship with the new nation.

Considering that trade with the colonies had been an important sector of the British economy before the war, it was reasonable for the king to wish for a friendly relationship with the USA and the resumption of trade instead of stirring up trouble with the USA by, for example, having John Adams arrested for his previous acts of treason.

So if the king managed to restrain himself from arresting John Adams for treason, what satisfaction would he get from arresting little known supporters of the American Revolution? So if it was safe for John Adams to be in London in 1785, what would Angelica Schuyler, who I never heard of before today, have to fear?

how could she be safe after travelling to London.

I believe this was a period when European nations did not much restrict the travel of their citizens, warfare was mostly conducted between armies & navies. Those civilians who could afford to travel were not necessarily treated as enemies or as combatants.

As Wikipedia reports:

After a brief visit to New York in 1785, the family sailed for England, taking up residence in London. Now the wife of a very wealthy man, Angelica entered a fashionable social circle that included the Prince of Wales (later King George IV), Whig party leader Charles James Fox, and playwright Richard Brinsley Sheridan. She also befriended and sponsored the émigré American painter John Trumbull, whose works included some of the most famous portraits of the American Revolutionary War era. Artists Richard and Maria Cosway also numbered among her close acquaintances in Europe.

Why did George III not decide to [take] revenge on her

I doubt he held her responsible for any personal injury. She was the wife of a British trader who became a member of parliament.