I am cutting it kinda close here!

Solution 1:

In the www.phrases.org.uk Bulletin Board forum, this phrase has been explained as a formation by analogy with cutting pieces out of cloth:

”Cutting it close” often occurs in the context of working against a deadline. ... To cut it close means to not leave enough margin. My guess is it’s by analogy with cutting pieces out of cloth in garment making. [R. Berg]
I thought it had its origins in tailoring as well, or carpentry, where if you cut too close to the line you've drawn you leave no margin for error--as in measure twice, cut once. Cut too close to the line and your work is wasted. [Markitos]

Wiktionary defines the phrase slightly more broadly than working against time, but offers no etymology:

cut it close
(idiomatic) To judge or finish something close to its limit.
Driving 300 miles on one tank of gas was cutting it close, but we arrived safely.

Note, the phrase “I am getting late” is not idiomatic English. In AE, one might instead say “I don't have much time”, or in BE, “I haven't much time”. Phrases like “It's getting late” or “I'm running late” might also be heard.

OED1 has several pages of senses of cut and phrases that begin with cut, but I've found neither cut it fine nor cut it close there, and little etymological information about the phrases. Entries “16. To cross (a line); expressing motion” and “17. To cross, to pass straight through or across; exp. cut over, cut across” seem suggestive of the origin of cut it close as used in regard to time or route planning.

Solution 2:

According to this "Useful english dictionary" page...

cut it close (chiefly US) (or chiefly Brit cut it fine)
to almost not be able to do something : to almost fail, lose, etc.

I was almost going to dismiss that source (if they don't know to capitalise English, how much should I trust them on anything else?). But comparing NGrams for British and American corpora, I find that in fact they're quite right.

OED places cut it fine in its section containing sub-senses of to cut = to carve (meat, etc.), wherein they also mention the slang usage to cut it too fat = to ‘come it strong’ = to overdo a thing, as used by Dickens a couple of centuries ago. Given that background, I don't think there's much point in attempting to hypothesise some original significance to the specific verb to cut - in this idiomatic context it probably always just meant any or all of assess, judge, carry out, perform, present, etc.