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.
So, one wouldn't say 'it's two past one' but could say 'it's one [o'clock]' or 'it's just gone one [o'clock]' in my dialect (Southern British English). I think from the opposite side I'd say 'almost' as in 'It's twenty-six past one' would be 'it's almost half (past) one'.
This is less common with a digital clock where its easier to just read the number than rounding it.
– Neil Tarrant May 09 '23 at 07:30