I'm not sure why the fuse died, I based the rating on the numbers on the label in your photograph but perhaps they showed average power and not peak if a discharged battery was attached. If that was the case, you could safely increase the fuse rating to say 2.5A without making it unsafe.
It is normal for the voltage to be higher from the charger than the battery. The reason is that in order to put charge into the battery you have to 'force feed' it a higher voltage than the cells themselves would normally hold. If you think of it the other way around, if the battery had higher voltage than the charger it would discharge into the charger rather than the other way around. I'm guessing the cells inside your battery pack are Nickel Cadmium ones as used in most power tools. These are charged with a constant current, in other words the voltage from the charger is allowed to change so that it is only higher than the battery enough to allow a fixed amount of current flow. As the battery charges up, it's voltage increases and the charger also increases it's voltage so it is always just a bit higher and can therefore keep 'pushing' current out.
It is possible, in fact quite likely, that if your battery is more than a year old or has had a lot of use, one or more of it's internal cells may have gone short circuit. When this happens, the battery loses capacity and the charging current may increase, that could be the cause of your fuse blowing. It isn't easy to confirm this unless you can open the battery casing. Usually it will contain several cells joined in a chain, each cell gives 1.25V (if it is good) so for a 12V battery you probably have 10 cells in the chain. What you do is charge it up for an hour or so, then unplug it from the charger and leave it for a few hours. Then using your testmeter you measure the voltage across each cell in turn, a fully charged one will show a little over 1.25V and ones that are failing will have dropped below 1.25V. Most faulty cells will have dropped to almost zero.
Be very careful if you try to open the battery pack. If you accidentally short out one or more good cells a huge current could flow and cause overheating or even an explosion.
Brian.