As a native Londoner, here is what I am most likely to say (and hear) at those times (given below in 24 hour format):
1302: "Two minutes past one." "It's just gone one."
1305: "Five past one."
1315: "Quarter past one."
1320: "It's twenty past."
If the hour can be inferred, it's often dropped. If people are asking because they need to catch a train at a particular time, or waiting for a meeting to start, the assumption is they're asking about the minutes, rather than the hour.
1325: "It's twenty-five past (one)."
1326: "It's just gone twenty-five past (one)."
Unless it's a circumstance where the listener needs the exact time, we're likely to approximate to the nearest 5 minutes.
1326, 1327: "about twenty five past". 1328-1329: "nearly half past"
1329: "It's nearly half-past (one)."
1330: "It's half past one on the dot." "It's half one exactly."
0934: "It's nearly twenty five to (ten)."
Anything other than the 5 minute increments of 25,20,15,10,5 feel strange to say. "It's twenty-six to ten." would be very rare to hear, much more likely to be exact for that and say "It's nine thirty-four."
0945: "It's quarter to ten."
We drop the "a" from "It's a quarter to ten."
"Quarter to" and "quarter past" feel like their own times.
0959: "It's nearly ten (o'clock)."
Someone reading a digital clock will likely tell you the reading, but with analogue clocks, conversationally, you are likely to get these more vague, conversational answers.