Slow switching up higher gears
It would have been slightly helpful if you told us from what front and back gear ratio (e.g. 2:3 or 2:4) the problem starts occurring. 

I had similar problems and it was even the case when I was changing to higher front gear but the cassettes (or clogs) did not change or they changed after a long time when it received external shocks from my ride. Did you check how the shifts are happening from 1(front):1(rear) onwards i.e. if you were to use the gear ratio from 1:1 to upwards? If you do this, then you will come to the point where you feel exactly from which gear ratio it is taking longer to switch cogs. Also, you might even find out once in a while changing to a higher gear actually does not change to the new cassette at all and gets stuck. One of the cable (front/rear derailleur) tension would not feel right to you.

For me it was some dirt and blockage in the front derailleur lever area (i.e. the allen key-based connnector with wire clenched around) which is universally near pedals. Usually, this is a minor problem and the way to fix it was to make sure first your gear ratio is 1:1. Then, loosen the allen key just enough so that the wire comes out. Now clean around the allen key are well enough and then use a little bit of wet lube around it to finalise. Then, put the wire back in and tighten the allen key as it was before. Reccheck the gear shifting from 1:1 upwards and see if the problem is solved. This could happen even to the rear derailleur too. The key problem originator is the derailleur lever (i.e. the allen key based moving area). What happens normally is that dirt and blockage occur there and the cable tension feels higher than it should normally feel. Also, please update your question by saying for how long and where (i.e. what type of tarmac) you have used your MTB until the problem started manifesting.

Also, I am assuming that you know NEVER to change both gears at the same time? If you don't know and have done this already, please don't do this again. For the investigation I mentioned above, make sure you change the gears down or up appropriately. 

Let us know :)....Don't forget to add the additional information to your original question :)

UPDATE

If these are of any help see This one

And also this

Hope this helps