There would have been no tactical use for an atomic weapon in 1944.
For those who claim that a nuclear weapon would be able to bust bunkers open your totally mistaken as the shockwave and overpressure at ground level is far less than conventional explosives which would have been far more effective in that role.
For those that assume that a major battle would be a good location you have to consider the blast and damage radius; from the bombs dropped on Japan we know that only about 26% of people were killed within 3mi of the epicenter of the blast, and that ~85% of all those killed were within an area about half a mile in radius.
Hence you've got about a 2km blast zone where it will be tactically useful.
Given that most battles ranged over tens of kilometers of front your hardly going to make a dent in even the most fierce battles of 1944. Thus while you'll certainly shock the troops in the region your not really dropping a battle winning wonder weapon.
Then we have to consider the effects of radiation. The weapon designers knew very well that the bombs would produce radioactive material, thus they knew they didn't really want to be moving into areas that had been bombed, at least not right after the blast.
Thus we get a situation like with the mines on the Somme in the Great War where having detonated you massive explosion you have to wait before moving forward giving the enemy time to fall back and create a new defensive line.
Because of the devastating nature of the weapon non of the allied countries (occupied or not) are going to want that weapon dropped on their land hence they are going to press for it to be dropped on Germany. Dropping the weapon here isn't really tactical since the battles here are yet to come for the most part.
We have to remember that there was an opinion to invite 3rd parties and deligations from the axis to watch the Trinity(?) blast to let the Axis know that the Allies had built the Atomic Bomb and that within a few months that was going to be dropped on their forces or cities. The idea being to let the axis commander know the war was lost and have them give up without ever having to use the weapon in anger.
If the Allies had the Atomic Bomb in 1944 that is the most likely course of action rather than actually dropping the bomb on Germany, and even if they had dropped it on Germany the way that the Nazis could use the propaganda of the Allies vapourising a city means that's the last place the Allies would want to drop the weapon.
Hence Berlin or any other German city is not an opition to demonstrate the weapon. This was an actual decision made by Allied high command earlier in the war (however it got changed later after VE day). Thus if the weapon was dropped in anger it would likely be some small millitary target away from populated areas (but close enough to be visable) with the attack pre-announced/or credited as soon as it happened, in order that it wouldn't give the Germans a propaganda victory and that the weapon would have the desired effect...shock...shock being much less if it is used in an actual battle and found to not be all that effective insuring the enemy doesn't fear its tactical use.
For these reasons,
there is no tactical use for an atomic weapon in 1944.
----------------------------------------------------------------------------
The only weapon of mass destruction which has the potential to be used on a tactical level is toxic gas.
Fuel Air Bombs?
EMP?
Radiological (i.e. Neutron Bomb)?
Herbicide?
I think there are some other catagories of weapon
. It's a good think that for the most part nations have stayed clear of weapons that are harmful to the general enviroment or cause indescriminate damage to large areas, however that doesn't mean that there are not tactical uses for such weapons. For instance a 'defoiliating agent' could clear an area of forest prior to an attack to deny defenders cover, or a radiological weapon could kill troops with a lethal exposure before the battle so they are dying off as your forces begin the attack.
Anyhow, that's an aside.