What does "uggin’ bumplies" mean?

Solution 1:

The phrase "uggin' bumplies" is most likely an attempt at a spoonerism for "bumpin' uglies", which is an old euphemism for engaging in intercourse.

Sources for "bumping uglies":
Online Slang Dictionary, ~1997
Urban Dictionary, ~2003
Yahoo Answers, ~2010
Reddit, ~2017
The Free Dictionary

Solution 2:

Without taking this question too far afield, I think it may be worth noting that "bump" as a slang term for copulation has been in use for quite some time. From J.E. Lighter, Random House Historical Dictionary of American Slang (1994):

bump v 1. to engage in copulation {with}. [First two cited occurrences:] 1669 New Academy of Complements 257: I'z bump thee quoth he..../ How lik'st it quoth he, well Thomas quoth she. 1921 McAlmon Hasty Bunch 187: He may be getting too much bumping from that widow Brown he knows...or just getting stupid from sitting around on his ass so much.

The more specific phrase "bump uglies" has been around for a while, too. Jonathon Green, Chambers Slang Dictionary (2008) offers this brief entry:

bump uglies v. (also bump nasties) {S[tandard] E[nglish] bump + ugly (bodies)} {1990s+} (US black/teen) to have sexual intercourse.

And Tom Dalzell & Terry Victor, The New Partridge Dictionary of Slang and Unconventional English (2006) notes a slightly earlier occurene:

bump uglies to have sex US [First cited occurrence:] And Tango adds a phrase to the popular lexicon when Sly's Tango asks Russell's Cash, "Did you bump uglies with my sister?"—USA Today, p. 7D, 22nd December 1989

A Google Books search moves the earliest documented occurrence back another few years. From a review by Jack Kroll of the film Sweet Dreams (1985) in an unidentified 1985 issue of Newsweek:

At the beginning she chuckles at the macho come-on of Charlie (Ed Harris) at the bar where she's singing. Scorning this seemingly dumb stud, she's delighted with his angry retort: "If I just want to bump uglies with somebody, I got plenty of places to go."

Sure enough, the Internet Movie Database reports this exchange in a conversation between Charlie Dick (Harris) and Patsy Cline (Jessica Lange) in Sweet Dreams:

Charlie Dick: I want to get to know you better.

Patsy Cline: Ok, now what does that mean?

Charlie Dick: Means I want to get to know you better.

Patsy Cline: See, I figure when you say you want to get to know me better — what you really mean is you want a ten minute screw in the back seat of your car.

Charlie Dick: Son of a bitch. You must think that thing between your legs is lined with gold. I can get tail any time I want, I don't have to come crawling after some mean mouthed woman who got a cob crossways. Hell, if I just want to bump uglies with somebody, I got plenty of places to go for that.

Patsy Cline: [laughing] Bump uglies?

Charlie Dick: That's right.

Patsy Cline: My lord Charlie, what a charming expression.

Sweet Dreams notwithstanding, my first exposure to the term (in slightly modified form) was from this excerpt in Edward Albee, The American Dream (1961):

MOMMY. Well, she's right. You can't live off people. I can live off you, because I married you. And aren't you lucky all I brought was Grandma. A lot of women I know would have brought their whole families to live off you. All I brought was Grandma. Grandma is all the family I have.

DADDY. I feel very fortunate.

MOMMY. You should. I have a right to live off of you because I married you, and because I used to let you get on top of me and bump your uglies; and I have a right to all your money when you die. And when you do, Grandma and I can live by ourselves ... if she's still here. Unless you have her put away in a nursing home.

There is, consequently, a fairly long tradition of "bumping uglies" in U.S. English slang to serve as a foundation to make the quasi-Spoonerism "uggin' bumplies" recognizable.