Well, this probably sounds gross and I'm kind of reluctant to say it, but I'm able to blow the snot out of my nose without hitting myself or my bike.
I try to only do it when there's no one around, but occasionally that isn't possible.
Basically, I turn my head so my nostrils are towards the ground and exhale forcibly through my nose.
I'm not sure it will work for everyone--perhaps I have remarkably straight nasal passages?
bicycles.stackexchange.com/users/2585/ananka
There's no great way.
The main problem is that inhaled cold air causes an automatic reaction in the nasal passages to create more mucous.
And to a degree simply having cold air blowing against the face around the sinuses
creates the problem.
Some people will experience this worse than others.
You can try using some sort of scarf or maybe a surgical mask over your nose to sort of warm the air going in.
The problem with this, of course, is that if your nose is running you can't wipe it without removing the scarf/mask.
Or you can use various drugs (OTC decongestants and nasal sprays) to try to control it.
bicycles.stackexchange.com/users/1584/Daniel R Hicks
Yeesh!
Why stop?
Spit down and to the side.
Keep going.
Main thing is to avoid spitting on the cyclist(s) behind you.
I really don't know why this is a problem.
Common sense usually works well. 2 Common Sense Rules Don't spit or blow snot on the folks behind you.
(Such as a pace line or other commuters) Avoid spitting or blowing snot on public places such as sidewalks or bike paths, etc.
On to peeing, I guess...
Out in the country, find a tree and go behind it...
in the city, find a bathroom at Starbucks or whatnot..
bicycles.stackexchange.com/users/None/
Keep your mouth closed when you breathe.
Holding the tongue against the roof of the mouth can make it easier to keep those passages warm and humid while you breathe through your nose.
bicycles.stackexchange.com/users/1097/ChrisW
Fisherman's friend or some other strong pepermint candy is a common remedy against such annoyances.
I like Fisherman's cause
it lasts quite long in the mouth - takes me 10 minutes to "eat" it.
bicycles.stackexchange.com/users/2794/Rook
Blow it away out of your nose while riding, to the side; Carry a cloth (handkerchief) to wipe your nose thereafter, so that it doesn't continue to be uncomfortable; If you come to a full stop, clean as much as you can with the two previous manoeuvers, so as to increase mucus-free time.
bicycles.stackexchange.com/users/2355/heltonbiker
I guess I'm on my own not wanting to see people, whether pedestrians or cyclists, flobbing out phlegm.
Take one of @Idigas' Fisherman's friends and use old school cotton hankies.
Even one-handed should be fine in many places (possibly not whilst running a red).
(Also not good is the sound of phlegm snorting.)
bicycles.stackexchange.com/users/128/Tom Hawtin - tackline
I have the same problem, which I solve by blowing to the side, and then wiping my nose on my shirt sleeve.
When my wife found out what I was doing, she crafted a velcro "snot rag" that wraps around my forearm.
Works well and easier to launder.
bicycles.stackexchange.com/users/2914/bluehole
A lot of gloves designed for running or biking have a little bit of fabric on the index and thumb specifically for wiping snot off.
You may want to consider using one of these, such as this one .
bicycles.stackexchange.com/users/8219/Batman
Now that the weather's cold, my nose gets plugged up after about a mile and inevitably some runs down my throat and I have to pull over and spit. I feel like stopping all the time is probably confusing to the cars behind me and seek a better solution. What's the best (safest?) way to deal with this?
Runny nose while biking
bicycles.stackexchange.com
<winter>
bicycles.stackexchange.com/questions/6779
bicycles.stackexchange.com/users/2050/fbo
Cold air when biking can cause mucus to build up, There is no great way to get rid of it but there are plenty tips for dealing with the issue.  A scarf or mask may keep the nasal passages warmer and therefore reduce the accumulation. Over-the-counter medication may help. Blowing the snot out of your nose as you are riding, obviously trying to avoid it landing on yourself, your bike, other cyclist or pedestrians. Running gloves usually have a special piece of material on them to deal with this issue or you can make a wrist band to use for wiping your nose. FInally, Fishermens friends are recommended for keeping nasal passages clear.  This is a common problem that affects some more than others, and cannot be fully avoided. It can be caused by inhaling cold air, so you can close or cover your mouth when breathing, take decongestents or use Fisherman's Friend to help reduce this, or carry a cloth to wipe your nose. Cycling gloves often have some fabric designed for this purpose too. Some people blow snot down and to the side, but other road users can find this disgusting. Use caution to avoid hitting people or footpaths. 
There is no good way to avoid this, but it can be caused by inhaling cold air. You can close your mouth, take decongestents or use Fisherman's Friend to help reduce this, or carry a cloth to wipe your nose. Cycling gloves often have some fabric designed for this purpose too. Some cyclists blow 'snot rockets', but this can be annoying to other road users and should definitely be done in a way that avoids hitting them or landing on footpaths.