I would totally mess around with facts to make the story work. I wouldn't move Scotland to the equator or put cell phones in the Nixon administration, but I wouldn't have a problem giving someone famous a previously unknown child, or adding an office building in downtown Denver, adding a secret passageway in The Chrysler Building, making up a national security protocol that never existed, or creating towns where none exist -- just to name a few.
Fictitious towns in real states or counties is quite common, as is creating a corporation or product or celebrity that doesn't (or didn't in the relevant time period) exist, even though the novel is set in the real world.
The reasons for doing this are obvious. You don't want to get sued for writing about a real person or corporation in a possibly inaccurate or defamatory way, so you make one up who may or may not bear a superficial resemblance to someone or something real. Or maybe you need a corporation or person who does something that a real one could have done but didn't.
As for towns, if I wanted to write a story set in a small, Midwestern town in the 50s, it makes sense for me to make my town up. Even if I do some careful research into real small, Midwestern towns from that era (a good idea) and base it loosely or more tightly on a real one, there's always going to be the issue of inaccuracies creeping in, or fictionalizations for the sake of the story tossing some people out.
Imagine someone reading your entirely fictional but set in a real place story and thinking, "What? I went to Mill Creek High in Illinois in 1954, and
I was the prom queen, not this imposter! Or wait, is this person supposed to be me? But I didn't have a secret abortion! OMG, I'm gonna sue!"
And of course, there's room for speculating about things that happened between the cracks of known history. Maybe some long-dead public figure from the 19th century who was notorious for his philanderings really did have a bastard that no one knew about. Maybe he had dozens. Any one of those possible bastards could be your character.
Changing the names of real people and places to protect their identities is another thing that's done fairly often. James Herriot (Alf Wight) created his famous semi-autobiographical veterinary stories from his own experiences and from those of some of his real-life colleagues, but he changed the names of many of the locations and people. He also rolled the time back a bit and altered some facts here and there, so it ended up being a work of fiction
based on his experiences. He did not, to my knowledge, make up any information about animal diseases, misrepresent the level of medical knowledge or the medical practices that were prevalent in the time in which his tales were set, however.
That's the line I'm talking about. It's one thing to massage the facts to make them work for your story, and it's another to club them to death and string them up.