The first time Mycroft met Doctor John Hamish Watson face to face he accused the man of trusting 'one, Sherlock Holmes' very quickly, for which he had his own reasons. However, in the end Mycroft began trusting John quite fast for someone in his position.
Sherlock explained with a little distain that his brother is the British Government, when he isn't too busy being the British Secret Service or the CIA on a freelance basis. Something that I'm sure only a small number of people are privy to.
When this was said outside the cabbie crime scene that night this was where the government worker could deny it. You could claim that there wasn't much point arguing when the actuality was out there. Or you could instead see it as Mycroft understanding that this was an important relationship that should be invested in.
John had already proved that he couldn't be bought, that he could be very loyal, very quickly to a man who many others avoided getting close to. Add to this the very likely possibility that Mycroft had already had worked out that this ex-soilder had also just saved Sherlock's life.
John Watson could have been a force for good or bad on the consulting detective. Good if he continued to support and protect him or bad if he was always this fast to jump in intensely or many worse things.
Whatever the outcome, Mycroft was willing to give him a chance and trust the man with some important information on the first day they'd met. It paid off to no end.