The use of "gone" in this manner is British slang; you would never hear an American use it this way.
Collins Dictionary gives the third definition of "gone" as follows:
If you say it is gone a particular time, you mean it is later than that time.
[Example:] It was just gone 7 o'clock this evening when I finished.
In actual use, though, there is a bit more to it. A speaker uses "gone" in this manner to convey the idea of where one stands (what time it is) in relation to larger block of time or upcoming event.
Example: She is six months gone.
Here we have a woman who is pregnant. Her pregnancy will last 9 months. She has been pregnant for 6 months and she will be pregnant for 3 more months. All of these ideas are part of the statement.
Example: It's gone ten o'clock and you are still not in bed!
Here we have a child who is still awake at an hour past their ordinary bedtime. Although the previous posters are correct that this usage of "gone" frequently indicates concern or frustration, this is not always the case. This example could be a stern warning on a school night from a parent or an affectionate comment from a visiting aunt.
Using the examples in your original question:
Example: It was gone ten o'clock by the time they arrived.
Most likely, here we have someone who has been waiting longer than they expected to wait. It may be as simple as that, or it could also indicate that the speaker was inconvenienced or missed out on something because of the late arrival.
But, this example can also mean nothing more than a plain, neutral "It was after ten o'clock when they arrived."
Example: It's gone six already.
Here we have have someone who is concerned about it being six o'clock because of a timing concern. It could be something like: It's gone six already and, although I've been working steadily, I'm worried I won't finish my paper by the deadline. Or: It's gone six already and if the cab doesn't show up in the next few minutes, we will miss our flight.
This example could also have a slightly more positive connotation of surprise instead of concern or frustration. You are having coffee with a new friend and chatting away, when one of you says, "It's gone six already!" meaning that you were so engrossed in the conversation that you did not notice the passing of time. This would usually be said as you were finishing coffee and moving to your next appointment; you usually wouldn't say this and stick around for another hour.