Robert,
Before discussing things further, I want to clear up a couple of things. The term "broad sword" is frequently used to refer to single handed swords used in the Middle Ages, but it's actually an erroneous and non-historical term. Because ARMA members are students of the sword as well as of historic European martial arts, we try to avoid using the term "broad sword" since it was not one used by medieval people. They would have simply called the weapon a "sword". If you want to be more specific than that about the type of sword you are referring to, referring to the single handed sword will make it clear what you mean. Also, if you're fighting in plate, you would almost certainly not be using a single handed sword. Since plate makes the use of a shield redundant, a long sword or great sword with enough room for two hands to grip the weapon would have been used.
http://www.thearma.org/essays/broadsword.htm Secondly, depending upon what you're referring to, war hammers are not that heavy either. As you can see from this replica by Arms & Armor of Minnesota, historical war hammers were typically not that long and the metal heads were not that large. So I think that it is unlikely that such a hammer would fatigue one significantly more rapidly than a sword.
http://www.armor.com/pole005.html You're question is a reasonable one, but difficult to answer in concrete terms. Simply put, despite the factors you gave such as mild weather, the answer to your question remains "It depends". Remember, the ability to withstand fatigue varies from individual to individual. Also, factors like how much fighting one has to do (assuming a battle) can make a difference too.
That having been said, we know that fatigue was a serious factor in fights. In accounts of some of the combats between people found in essays in this website, I believe there are instances where men on either side were forced to stop and rest from fatigue. In the judicial duel between Jean de Carrouges and Jacques le Gris, it is indicated that the fighters would some times rest for a bit before attacking, despite the fact that the duel was fought in late December in France and the weather was cool. In his book on Hastings, Peter Poyntz Wright indicates that William's men probably stopped to rest after each of their failed attempts to take Senlac ridge. From what I've read, it was certainly not uncommon for fighting men to take rests that were mutally but wordlessly agreed upon even though they might be mortal enemies. So you're quite right that armour posed a serious fatigue factor, because we have accounts of people requiring breaks to rest. Just how long a particular fighter could go without needing a rest is up for debate however. It's one of those intangibles that is really impossible to answer.