The champion that has the highest speed gets the first turn.

If you do not get a turn, then your enemy had higher speed.

Most buffs require accuracy to land. You can counter this with a resist stat. If your enemies have high accuracy and you have low resist, the buffs will land easily.

And the other way around, if your enemies have a high resistance and you have a low accuracy, you will not be able to land any buffs.

HP does not have a lot to do with it.

Your champions have more speed and you still don't get a turn? No, definetly not. You don't know the speed stat of the opponent's team before the fight starts, and if your champs don't get a turn, you obviously have less speed than they have.

Do you have boots with speed as main stat on your champs, at least on your speed booster? What is the actual speed of this champ?

"i  still  don't  get  a  turn  even  with  teams  with  half  the  HP"

"teams with half the HP" - are probably speed-nukes. With a speed-leadup and overal speedy.

Such nuke of mine in arena "silver 2, 3" has lead with speed 204+10%. And it is kind of mediocre...

Yes I'm afraid team power counts for very little in the arena, it is the composition of the enemy team you need to be able to recognise, I'm guessing you're not in a high tier but if your opponent has speed you can recognise that by the lead, High Katun for example is an easy one to get and provides a speed boost, then they may have a debuffer/weake/def down going next, then you'll get nuked without having a go, once you can recognise the team composition you can pick better fights