As a Sniper, how can I deal with being ambushed by a Pyro?

Most likely, as a Sniper, you are far away from the front line... so not many people would come barging their way towards you. However, in the case that they do, you have a few choices:

1. Retreat. Backpeddle away as quickly as possible while dealing as much damage if you have any secondary damage dealing weapons like the SMG or the Cleaner's Carvine. Lead them to your team who can finish him off better than you. Do not use your Primary, because to use your primary deals very low DPS per second and without scope, it is very hard to land hits because of the low fire rate of any rifles.
2. Throw your Jarate at him, then use your Bushwacka to hit him to deal full crits. One or two hits from a full crit bushwacka should take out most of the class in the game.
3. If you are using Huntsman or the Fortified Compound, retreat first, and as you backpeddle, fire headshots at the enemy. A fully charged headshot from a Huntsman or Fortified Compound can kill all class except an overhealed heavy.

If you see Pyros charging at you, but has not yet reached you, do a quickscope to his face and another bodyshot will kill him. That will most probably take about a second, considering you don't miss.

If you are cornered, the weapon to have is the SMG and the Shahanshah. Use your SMG till you run below half your health, then use your Shahanshah to deal heavy blows and finish him off.

For Pyros, it is advisable that you do not use Huntsman or the Fortified Compound - they can easily air blast them away and send them back to you.

Hope I helped :)

The truth of the matter is that there's only one real way to counter this situation - avoid it entirley.

At medium-range, you can try to huntsman a pyro, you can try to get a shot off or jarate/bushwhacka him, you can even try your SMG if you're desperate, but the only surefire way to avoid getting cornered by a Pyro like this is to plan ahead, and keep yourself aware of your surroundings.

By this token, there are a few things you can do.

Do not stand in a corner, or anywhere where there's no possible escape route

Your first mistake was choosing a sniper location from which there was no escape. Always have a plan of retreat when sniping. Or for any class really.

Do not snipe from just one place

In keeping with the above, don't just snipe from one place and never move. The Killcam will ensure that you are hunted down and destroyed after one or two headshots. People love to take out a hard-scoped sniper, or just one that is always in the same spot. So switch up your positioning.

Stay behind your team

A sniper's best counter to a pyro is a class better-suited to countering pyros. You're terrible at taking on their short-range attacks, so stay behind everyone else in your team and in locations they'll be hard-pressed to reach. If your team is pressed back, be as pressed back as possible, and possibly consider switching from sniper (you aren't going to do your team a lot of good if the enemy is already up in your face).

Run! Run! Run!

There's no shame in beating a hasty retreat. Snipers are one of the weakest close-range classes out there. Only a few other classes are viable to take down at close-range with a quick Jarate/Bushwacka, and medium-range combat really isn't your forte either (Unless you're a Huntsman Sniper). Run, run, run away, and live to snipe another day.