The great focus of the nuclear weapons effort early on was on hitting Germany (using B-29s based out of Aldergrove in Northern Ireland); that really didn't change until October/November 1944. the actual change in priority from Germany to Japan isn't documented; we have to go by inference. in August 1944, the plan was blast Germany out of the war, in January 1945 it was blast Japan out of the war so something happened between those dates. Interestingly, the Ardennes Offensive didn't change much.
So, assuming that the Allies did have nuclear weapons in August 1944, there is no doubt that Germany would be the target and that the attack would be carried out by B-29s. The only question would be whether the weapons were used as they came off the production lines or whether a number were assembled and used in a major opening blow. A lot of that will depend on personalities. Also, at that time, it was believed that Germany had an atomic bomb effort of its own and was roughly level with the US. That belief died between October 1944 and January 1945 as well but in August 1944 it was seriously accepted. That would have added pressure to the "use now" school. On the other hand, the proponents of "assemble a sledgehammer" would point out that using just one bomb would effectively tell the Germans (a) it could be done and (b) how to do it. It could well be argued that using one or two nuclear devices in August 1944 would guarantee the Germans doing so in December 1944.
Another big question; if the US knows that nuclear weapons are coming in August 1944 (they'd probably be certain of that in May at the latest) would Overlord happen? Why invade when the power exists to reduce Germany to a radioactive slagheap? Would the bomber offensive have picked up steam the way it did? Why use conventional bombing by B-17s and B-24s to slowly reduce German cities to rubble when single B-29s can do so overnight? I can see this being a really agonized and frank debate. Yet another point is production capacity. A nuclear device may have been available in July 1944 but is it a production item? By December 1945, the planned production rate of nuclear devices was going to be seven Model 1561s and one Mark One per month (it never got there). In the revised timeline, would that production rate have been achieved in December 1944? What would the production rate have been in 1945? I've never seen an estimate for 1946 production assuming the war had continued although plans for Operation Coronet in March 1946 included the use of nuclear weapons tactically (and prolifically) so we can assume the production rate would have increased significantly. IIRC the first day of Coronet would have seen 13 nuclear devices initiated over targets in the Kwanto. I would say that is a likely minimum for an attack on Germany.
Targets? Dresden in my opinion is a rock-hard certainty. Given its industrial importance, it's odd it wasn't bombed in 1944 when it got to be within effective range of allied bombers and I believe that it was being kept as a sample target do see what a device would do to an undamaged target. As for the rest, industrial facilities, mostly oil and aircraft. Nobody will waste time trying to get the German leadership or waste scarce devices on "political" targets. One thought, assuming the conventional bombing offensive doesn't build up because early nuclear devices are coming, German industry will not be dispersed so there are nice concentrated targets to incinerate.
There's another factor here; Japan. Let's suppose, for the sake of argument, that the leadership wakes up one morning and at the situation briefing get told what's happening in the Pacific. then the following exchange happens.
"Well, what is the news from the Germans?"
"What Germans?"
There was a peace party in Japan then although it was weak, powerless and ineffectual until after Hiroshima and Nagasaki. Would they have pointed to the smoking hole where Germany used to be and say "That's us in a year's time. Give it up now you damned fools." and made it stick?
In short, nuclear weapons in August 1944 is very bad for Germany, has the potential of being extremely bad or very good for Japan.