A swimmer swims, a baker bakes, and a painter paints. Who picks up? A picker up? A pick upper? A picker upper? An up-picker? Logically it seems like “picker up” should be correct, but it sounds odd to me.
“My wife is a painter and that guy over there is a picker up” :dubious:
If something is what you put in, you call it input. What is put out is the output. That which is taken up is the uptake. The person who takes you underground is the undertaker. 
I think up-picker would be the best neologism here, but I’m not sure this has a factual answer. There do seem to be a fair number of words in this mode: outfitter, undertaker, upseller, etc.
Phrasal verbs (I think they’re called) like “pick-up”, “chip-in” etc. are notoriously awkward to put into that ‘doer’ form.
If you rely on the classic Bounty paper towels commercials, it would be “picker-upper”. It’s like asking who “swims up” or “bakes up” or “paints up” or “calls up”. A swimmer swims. A picker picks. Someone would be a “caller” whether they ‘call up’, ‘call on’, or ‘call out’. I don’t claim this is the factual answer but I would think “picker” applies whether one ‘picks up’ ‘picks on’ or ‘picks out’. A cleaner ‘cleans up’ but they wouldn’t be called an up-cleaner or cleaner-upper (Bounty paper towel terminology notwithstanding).
The janitor.
To pick up is not the same as to pick. While “up-picker” makes logical sense, no one uses it and it thus sounds contrived. “Picker upper” is quite informal and the “-ers” seem redundant, but this is how it has been done for decades, and people understand it.
Wait a minute – you mean he’s trying to pick’er up? Cheeky bloke, ain’t he?
You can’t fool me. Everyone knows janitors janit.
There are “pickers” who literally to pick stuff up. Some pick fruits and vegetables. Some pick out items in a warehouse to fill an order.
Or, am I being whooshed?
Uppickerer.
Live and learn; I thought Autolycus (not the ladle-abuser) was a “picker-up of unconsidered trifles”, but it turns out he’s a “snapper-up”. I got nothin’, in that case.
A pick-up artist? 
Mom.
Though I’d go with picker-upper for familiarity, even though pick-upper seems more correct to me.
The latter is also used. In fact, if I were to come up with a rule for phrasal verbs, it would be that one. The verb is considered to be a unit, so you add the -er to the end of the phrase itself.
I know that, as kids, my sister and I played “pick-upper/put-upper” for cleaning. One of us would pick up stuff, and then put it in a pile for where it goes. The other person would, every once in a while, take the pile to the room where they belonged and put it all up. As an adult, doing only one job gets boring, so I play both parts.
If it were a unit, you’d expect inflections to happen to the end of the second word but you’ll find that most people prefer threw up to throw upped, picked up to pick upped and jacked up to jack upped.
I think it’s called nit-picker. 
I heard a little (English) boy on TV the other day express a wish to become a “tree-chopper down”.