Hagerty's a good real-world figure for this but it does mean Christian Communism couldn't start until 1895 at earliest (assuming he starts immediately). That's near the end of the Third Great Awakening.
The Communist Manifesto reaches the US in 1872 so if that starts to catch on with some priests, you can have Christian Communism be part of the Awakening around then but then who starts it?
If 1872 or close to, it could end up being a big deal. By the time of the Russian Revolution(s), this form of communism will be 45 years old; it'll be an established part of American politics and can go "well that's atheist communists for you". It'll probably quite popular too, based on the success of Great Awakening reforms and the working conditions of the 19th & early 20th century. Other communists aren't going to like this at all but if Christian Communism catches on first, what are they gonna do? This could be good for the Socialist Party of America, deflecting a lot of the backlash and crackdowns they got (now if only they can stop internal fights).
The Awakening also liked missionaries, so you'll have Christian Communists going abroad - I doubt they'll be popular with governments overseas (or foreign communists) but they could start spreading the beliefs. Maybe they'll be allowed in to Europe by governments that think "ahahaha, go fight among yourselves commies".
Hagerty's going to dive into this headfirst and, by the sound of his work ethic and rhetoric, become a big man in the movement.
(What does this do for race relations? Communism
should be an anti-racist ideology but will that be downplayed, left out, or just outright ignored by Christian Communists when nobody's looking?)
If the Great Depression still happens as in our timeline, here's a pre-existing ideology with 56 years of weight behind it, lots of speakers and ideas that will resonate and doesn't require you to turn on the church. Its time will have truly come. And it'll explode right in time for two congressional elections, two senate elections, some gubernational elections, and a presidential election. Christian Communists are going to get in power and other men will get in power by appealing to Christian Communists. If the Socialist Party survived,
Norman Thomas - who is going to be a CC himself in this timeline - could become president (and likely already has been a New York State Senator after 1926). A Socialist government, or coalition of CC politicians, goes on to bring in socialist policies to bring America out of the Great Depression.
What happens after that depends on how good the Thomas administration is, how many competent members it would have, if the Socialists and other CCs can avoid ripping into each other for not being socialist enough, how foreign powers respond to the US being a bunch of bally commies etc etc etc. Either way, their victory will embolden Christian Communists and socialist parties in other countries. If Thomas does a good job or at least can be
presented as doing a good job, the Labour Party makes big gains at the 1935 general election - possibly wins, giving us an Atlee government ten years early. (This one requires a
big gain for Labour but the USA voting Socialist would be a big deal. A downside here is that it would delay our rearmament and make us weaker if/when war breaks out with Germany) The Thomas government
may support the Republicans in Spain with money and arms (but not likely to send troops), as antimilitarism may seem like a bad policy when it's fellow lefties getting bombed instead of a war between European imperialists, which would alter the course of the Spanish Civil War. (If he loses the 1936 election and the fascists are still around then, however, it'll get right back on track)
If Thomas wins a second election, do we get a Second World War as we know it? If Hitler thinks the United States might intervene - even just with guns and money - is he going to want to risk conquering Poland? (Czechslovakia, he needs to risk because it gets him Sudetenland
and a way to march into the USSR) Will he even be able to invade the USSR which, even if it's an atheist communist dictatorship, will still be a communist country?