Firstly: 

It is permissible to wipe over the army boots, because they come under the same ruling as the khuff (leather slipper). 

Shaykh al-Fawzaan (may Allah preserve him) was asked: Is it permissible to pray in army boots? How should one do wudoo’ when wearing them? Is there a specific time limit for that? 

He replied: It is permissible to pray in army boots - i.e., the khuff - if it is clean and pure and there is no najaasah (impurity) on them. 

It is permissible to wipe over them when doing wudoo’ if they cover the foot completely and come to the ankle and are attached firmly to the foot and were put on when the person had wudoo’. 

The way in which wiping is done is to put the wet fingers on the ends of the toes, then bring them up to the shins. 

The time for wiping over them is one day and night for one who is not travelling, and three days and nights for the one who is travelling. This is a concession which is confirmed in mutawaatir reports of the Sunnah of the Prophet (blessings and peace of Allah be upon him) and no one denies it except an innovator. That time period, according to the correct view, begins with the first time one wipes over them after putting them on. And Allah knows best. End quote from al-Muntaqa, 2/54. 

They also the answer to question number 69793. 

Secondly: 

One of the conditions of wiping over the khuffayn or whatever comes under the same ruling, such as army boots, is that they should have been put on when one was in a state of complete purity. This has been discussed in the answer to question number 9640. 

If you still had your first wudoo’ in which you washed your feet, and put on the army boots after taking off the sport boots, it is permissible for you to wipe over them, because it is true that you put them on when in a state of complete purity. 

So you may wipe over the boots for one day and night if you are not travelling, and for three days and nights if you are travelling. The time starts from when you first wipe over them after breaking wudoo’. 

If you need to take off the boots, then you may wipe over the socks. In that case it will not matter if you take off the boots or trainers, because the ruling applies only to the socks. 

And Allah knows best.