The UK has a constitution, but it was not constructed systematically in the way of most modern written constitutions, and it can be extended when necessary. The abdication of Edward VIII caused some significant precedents to be set. There were no established procedures, because nothing like this had ever happened before.
His ministers and other influential people convinced him that he could not marry Wallis Simpson and retain public support. Modern standards of behaviour are rather different, so it may seem strange.
- The monarch is Head of the Church of England, which is still unwilling today to marry divorced people whose spouses are still alive. Therefore, he could not marry Simpson under the aegis of the church of which he was the figurehead.
- Many of the people who had met Simpson were very unimpressed by her. Being American was not a social advantage in upper-class England of the time, and many people thought she was after money and power, rather than loving the king. They seem to have been wrong about that, but it was the feeling at the time.
- The monarch needs public support. The British are well aware that the monarchy is an anachronism, and somewhat silly. If the monarchy is not popular, its contradictions in the modern day become much more obvious.
His renunciation of the throne was legal, in the sense that there was no law against it. It did not take legal effect until Parliament had passed His Majesty's Declaration of Abdication Act 1936. He actually continued to be monarch until that act came into effect, which happened when he gave Royal Assent to it, and thereby ceased to be monarch. All of this happened the day after he had renounced the throne: Parliament can act quickly when everyone agrees that's a good idea. Each realm of which he was King had to pass its own act to give effect to the abdication, which is why he was King of Ireland for a day longer than anywhere else. If any future UK monarch abdicates, the procedure will presumably be similar.
To retain the throne, he would have had to abandon his plans to marry Mrs Simpson, and he was unwilling to do that. He could have remained unmarried and kept her as a mistress, but she would not have had a social position along with him. He also had an implied duty to father an heir, and she was a bit old for that in 1936.
There was a covert objection to Edward VIII, among those who knew him. He had the maturity of a teenager, entirely self-centred and caring only for his own pleasures and desires. My source for this is the diaries of "Tommy" Lascelles who was Private Secretary to Edward VIII, George VI and Elizabeth II. George V had regarded Edward as poorly suited to being king and hoped that it would end up with "Bertie" (George VI) and his daughter Elizabeth, as actually happened.
The story that he was in favour of Nazi Germany has a distinct problem. If he was, why was he picked out of his exile in France and attached to the British Military Mission to France? It's very plausible that he was impressed by the glamour and displayed power of Nazism, but that's different from being a traitor.
Looking at the original version of the question, there are no established ways in the UK to persuade or pressure a monarch to abdicate. In the post-Victoria era, if a monarch lost public support completely, their choices would be to tough it out, which may well end the monarchy entirely, or to abdicate. Since Edward VIII was mainly interested in his own pleasures and desires, abdication was presumably the easier course for him.