There are two figures for this: the theoretical maximum if they had continued at wartime levels, and the actual production rate that occurred.
The conservative theoretical maximum was around 3 per month. This was just the Hanford production rate, which was around 18 kg of plutonium per month, which at ~6 kg per Fat Man bomb, corresponds to about 3 Fat Man bombs. (Technical note: the reactors could produce 200 g of plutonium from every ton of uranium metal put into them, and could process 30 a month, and there were 3 of them = 200 * 30 * 3 = 1,800 grams of Pu per month.)
This does not include the highly enriched uranium (HEU) method. In late 1945 the daily output of enriched uranium from Oak Ridge was about .8 kg/day. It takes 64 kg to make a Little Boy bomb, so that's one bomb every couple of months. By the end of 1945 however they had improved the process enough to raise the output to 1-2 kg/day. So that's one Little Boy bomb a month after December 1945, and 2 bombs per month after around March 1946.
If they were really looking to increase their output they could have also developed "composite" core bombs, which mixed high-enriched uranium and plutonium, in what they called the "combination" bombs at the time. They knew about this possibility and knew it would entail a great savings in fissile material, because you could use a little bit of Pu with a little bit of enriched uranium and have an implosion bomb that worked the same or even better than a pure Pu bomb. So you could stretch your Pu out that way, and get to use that uranium more efficiently.
The first generation "composite" bombs had about 2 kg of Pu to 5 kg of HEU in them. If Hanford produced 21 kg of Pu per month while Oak Ridge was producing, say, 30 kg of HEU per month, you'd get 6 "composite" bombs per month, plus 1 pure Pu bomb, plus 3 kg of Pu left over per month. So that's a pretty good output, but relies on developing a new bomb design (one they were working on during the war, but didn't actually proof-test until afterwards).
OK, that's the theoretical maximum output. But the reality was different, because the war ended in late 1945 and the Manhattan Project's status went into a kind of limbo for a few years. I have written about this here, but the gist is that:
With the war over, a lot of people went home
There was uncertainty about the US future nuclear policy stance
There was uncertainty about whether the Manhattan Projects sites would continue or not
There were technical problems at Hanford with the running of the reactors (graphite swelling due to prolonged neutron flux) which required running them at half-power until fixes could be discovered
New weapons design was essentially put on hold
So the result is that the number of actually produced bomb cores per month dropped dramatically, and the actually ready-to-use bombs was at or around zero through early 1947. Bomb production rate in 1945 was about 1.5 per month; in 1946 it dropped to around 0.4 per month; in 1947 it was up to 1.75 per month; by 1948 things were back on track again and it was as high as 6.5 per month. In the late 1940s, the upgrades to the bomb production system put into place allowed to stockpile to balloon upwards in the 1950s (thousands of bombs per year) but that took time to get working, especially since the Manhattan Project infrastructure had fallen apart.
Some of those infrastructure problems might have existed had the war continued (e.g. the Hanford swelling issue), though many were distinctly caused by the end of the war, and had the war gone on, it seems likely they would have gotten quicker fixes to them (e.g. developing the "composite" bomb approach to compensate for the plutonium shortage — as it was, they didn't actually test a "composite" bomb until 1948).
There are several "take-aways" here, including a discussion (linked above) about the importance of maintenance in big infrastructure projects, but I do like to bring this up whenever people say, "well, they only had two bombs, thus they couldn't have made a demonstration bombing." They had two bombs ready to use in early August 1945, but they had a production line that, if continued, would have produced many more bombs per month, with plenty to spare before the date set for the invasion of Kyushu (November 1945), certainly more before any set invasion of the Kanto plain (spring 1946). So they were not as cramped for time as it often gets made out to be. That is not to say that a demonstration would necessarily have been a good idea, or that they shouldn't have used the bombs as soon as they got them, but a lot of people have bought into this scarcity narrative that is not quite correct, or even worse believe that the promise to use more atomic bombs was a "bluff," when really they had more bombs coming down the pipeline (the next one was planned for August 17 or so).