I love alternate history scenarios; however, I just want to start writing. Should I do research or just start writing and go with it with some research along the way?
I think you need to do light research, unless you already know the answers.
For example, suppose you thwart the assassination of JFK (Stephen King already wrote that book). We know what happened (real history) when Lyndon B. Johnson took over, but you need a plausible theory of what happens if JFK remains President. I don't think you can just do anything, this topic has been speculated upon by experts including people close to JFK. If you impose your own invented personality on JFK, it will not come off as a plausible alternative history.
Of course the further in the past your change, the less people know about it, and the more leeway you have to impose an invented personality.
Likewise, as an alternative history, you can kill MORE people than JFK. What if Lyndon Johnson (riding in the car behind JFK) was assassinated also? That would make John McCormack (the Speaker of the House and next in the line of succession) President.
You would have more leeway then, McCormack is not a well-known political figure, but a little research on his war stance and such might lead to a very different Presidency. The advantage here is you don't have to figure out what JFK would have done, you can write a little more freely about President McCormack. He was a Democrat and publicly supported the New Deal, Great Society, civil rights, public education and health care, and advocated for the Vietnam War to fight communism.
My overall point here is that your story should be mostly fantasy, but an alternate history demands a few anchors in the real world, when you are dealing with real people that other real people remember, or can look up on Wikipedia. Otherwise your alternate history is not believable. If you make obvious mistakes, then well-read people (e.g. agents and publishers and reviewers) that do a little research and find obvious flaws in your real-history claims won't buy it or recommend it.
That does NOT mean you have to do all your research up front, you can still be a discovery writer and start writing, but you will have to stop frequently and resort to Wikipedia and Google when you are writing about real historical personalities. That doesn't have to take long, I read ten minutes about McCormack. He might prosecute Vietnam differently, but he would have tried to win it. And he would still have pushed JFK's civil rights legislation through and the Great Society of LBJ, perhaps better than LBJ.
Fantasy and Scifi I think you can get away with writing while doing no research. I don't believe that is true of this particular niche; alternative history.
I think you can minimize the research you need with the right strategy, namely eliminating the well-known people and resetting the field, so your own imagination takes the reins fairly early in the story. In the example of JFK, kill LBJ also. Then you can write about real people that were not as well known, and introduce your entirely fictional characters early; e.g. make McCormack's VP a fictional character designed by you to be a major player and source of additional conflict going forward; heck the fictional new VP can be one of your main characters.
It would depend on the scenario. I suggest you just write, let the changes to history that you make lead you and the reader to interesting places.
The movie Fatherland has an interesting scenario where World War II never happened and JFK was not assassinated. This led to a world with a thriving Germany ruled by the Nazi party and Hitler, the US President was none other than an older JFK.
Let your dominoes fall or not. Start writing and see where it leads. The ‘what if x never happened’ can be a compelling question to explore. Consequences radiate outward and the changes caused by one alteration can be considerable and far reaching. Sounds like fun.
If the era is far enough back in time that most readers might not be immediately aware of its history, you might want to research first.
I would do some research first.
But what some might call research, others would call fun reading. So, yeah, you should have a decent grasp on the real world history first. As well as a basic understanding of the technology or politics or cultural issues you want to change.
After that, it really depends on your personality and how you work. Some people (like me) like to research the hell out of something before starting. Then maybe write a bit and research some more. Others want to just start writing. As long as you're not going in completely blind (don't write a "what if" happening in WWI if you don't know the basics about WWI), any approach is fine.
You also need to know yourself. Are you someone who will research endlessly and never feel you understand it well enough to move forward? Just write already! Don't wait for perfection. Or are you someone who plunges ahead without a sense of what you're doing or why, and ends up with unusable pages? In that case, maybe step back and do some formal research first. It's okay to write for the practice alone (and most stuff can be reworked into something else), but after a while, you are gonna want to produce something you can use.