Knives are useful tools for cutting bagels and applying delicious spreads such as peanut butter or cream cheese, but luckily if you are missing this implement, you can still enjoy your bagel with 3 easy steps.
Rip your bagel into dunkable pieces.
Swipe them through the peanut butter or cream cheese.
Cram them into your gullet.
lifehacks.stackexchange.com/users/168/Sterno
If you can find some dental floss or thread, you can use that to cut the bagel.
Wrap it around the circumference (as in picture), then cross ends and pull through.
Bagel ready for the toaster.
lifehacks.stackexchange.com/users/6487/Debbie M.
Look around and see if you can spot a letter opener (old usage was "paper knife").
It's likely to be sharp enough, with patience, to cut the bagel, and it will surely work to spread your peanut butter and cream cheese.
Failing
that, a plastic fork from the lunch room can probably be used to split the bagel (push the tines in repeatedly along a line where you'd normally cut; when you've worked your way all around, the bagel should split easily along the perforations), and will also spread the toppings, after a fashion.
lifehacks.stackexchange.com/users/9830/Zeiss Ikon
find a piece of plastic like a telephone card or a credit card, abrade the edge against a rough surface like a stone or even bite little dents in it, or use any metal object, and use that to cut.
takes 2 minutes. find a magazine cover, glossy one, or a plastic film, put a book on the bread and use it as a knife, if it's abraded it will be faster. find a thin wire like a copper wire, abrade it, use it like a cheese cutter.
lifehacks.stackexchange.com/users/6443/aliential
Some toasters in the UK* have a warming rack that can be used to heat bagels, crumpets and the like.
The warming rack is a pop-up that allows you to perch your larger items above the slots, to warm the item with the rising heat.
Simple use the warming rack to warm your bagel, and then dip it using @Sterno's method.
*Apparently this also applies to most of Western Europe, but not to the USA.
Unfortunately the USA is where most bagels are consumed, which rather limits the usefulness of this answer!
lifehacks.stackexchange.com/users/11101/AndyT
use a pair of scissors, either as a defacto knife, or in their intended way.
give them a clean first!
lifehacks.stackexchange.com/users/8207/Adam
Take a bite of the bagel, then scoop your finger into the respective peanut butter or cream cheese, and place it into your mouth at the same time you have the bite of bagel in your mouth.
Mash up food in your mouth together.
It's a little bit involved, but it is satisfying doing things for yourself!
lifehacks.stackexchange.com/users/11665/Irony101
If you pinch the bagel with both hands, thumbs inside the hole and fingers around the circumference, then use enought force to break through the crust with your thumbs (those with long nails may find this easier), then pull gently apart, but not completely, you'll have separated a portion of the bagel close to the middle.
Rotate your fingers and thumb slightly and perform the process again.
Do this repeatedly and
soon you'll have two halves of the bagel, as though you'd cut them with a knife, though much more ragged.
They will now fit into the toaster.
For spreading, you can simply use your fingers, but if keeping your fingers clean is important or you can't wash them afterwards, use a napkin or paper towel wrapped around a finger or two to gather the spread and apply it to the bagel halves.
lifehacks.stackexchange.com/users/1801/Adam Davis
Your car keys might work in a pinch (for cutting anyhow).
Clean them thoroughly first, of course.
For spreading, a plastic spoon works just as well as a knife.
Failing that, you can always use your finger for spreading.
A bit messy, and unhygienic if others are sharing your cream cheese, but otherwise effective.
lifehacks.stackexchange.com/users/10020/Darrel Hoffman
What is wrong with you all?
Have you forgotten about MacGyver ?!
All it takes to solve this problem is one of each of the following: Paper clip Pen Rubber band
Once you have secured the above items, follow these steps: Unwind the paper clip and STAB
the bagel where you will begin cutting it.
Then proceed to saw your way through it and around it until you have a nice bagel cut in two.
Pop these two slices in the toaster.
Dismantle the pen, leaving only the long housing piece.
Wash this piece and dry it.
Once your bagel is done toasting, spread your desired spreads on the bagel using your clean spreader from the previous step.
Enjoy!
But wait...
what about the rubber band, you say?
To be honest, we'll have to ask MacGyver why we'd need a rubber band here!
Nonetheless, this is as MacGyver-like as it gets without the real MacGyver.
For the meantime, I guess you can just wear the rubber band around your wrist as a bracelet as you eat your tasty bagel.
P.S.
After you've devoured your meal, bend paper clip back in place, wash the pen housing piece and reassemble the pen, take off your rubber band bracelet, and return these three items to their places before anyone finds evidence of what took place and starts hailing you as the office hero!
lifehacks.stackexchange.com/users/11960/Pamela
I brought a bagel into the office for breakfast, but the only knife in the office sharp enough to cut bagels has gone missing. I tried to shove it into the toaster anyways, but it was too big. I've got some peanut butter and cream cheese, but no way to spread it, and I don't want to eat the bagel plain.  How can I go about eating this bagel with some spreads on it in order to add some flavour to it?
How can I eat my bagel when I lack a knife to cut it?
lifehacks.stackexchange.com
<food><substitute-tool><work>
lifehacks.stackexchange.com/questions/10622
lifehacks.stackexchange.com/users/165/Wipqozn
If you lack a knife to cut your bagel or apply spreads, you could simply rip up your bagel and dip the pieces in the peanut butter or cream cheese. Alternatively, you could simply try to cut the begal using either dental floss, a letter opener, an old credit card or very carefully with your fingers. Car keys may also suffice. As a last resort, you could simply eat your bagel whole and use your fingers to eat your spread at the same time. The options are to tear up the bagels (possibly heating them first over a toaster) and swipe the pieces through the cream cheese or peanut butter, or use dental floss, a paper cutter, scissors, car keys, your fingers or a bit of plastic such as a credit card to cut open the bagels and spread the filling with your finger directly or wrap your finger in a napkin first.
Tear up the bagels and swipe them through the peanut butter or cream cheese. Use dental floss or thread to cut the bagels in half. Use a letter opener or paper knife. Use a piece of plastic like a credit card, bite to make a sharp edge. Warm the bagels over the top of the toaster and then tear up and swipe through the spreads. Use a pair of scissors as a knife. Bite the bagel and use a finger to get the peanut butter or cream cheese into your mouth. Cut the bagel in half by pushing your thumbs through and gently pulling apart then use fingers or a paper towel to spread the filler. Cut the bagels with your car keys and spread the filling with a plastic spoon or your finger.