Following is an excerpt from Goblet of Fire :

‘Oh yes,’ said Mr Weasley, tucking the tickets safely into the back pocket of his jeans. ‘The Department of Magical Transportation had to fine a couple of people the other day for Apparating without a licence. It’s not easy, Apparition, and when it’s not done properly it can lead to nasty complications. This pair I’m talking about went and splinched themselves.’

Everyone around the table except Harry winced.

‘Er – splinched?’ said Harry.
‘They left half of themselves behind,’ said Mr Weasley, now spooning large amounts of treacle onto his porridge. ‘So, of course, they were stuck. Couldn’t move either way. Had to wait for the Accidental Magic Reversal Squad to sort them out. Meant a fair old bit of paperwork, I can tell you, what with the Muggles who spotted the body parts they’d left behind …’
Harry had a sudden vision of a pair of legs and an eyeball lying abandoned on the pavement of Privet Drive.
‘Were they OK?’ he asked, startled.

‘Oh yes,’ said Mr Weasley matter-of-factly. ‘But they got a heavy fine, and I don’t think they’ll be trying it again in a hurry. You don’t mess around with Apparition. There are plenty of adult wizards who don’t bother with it. Prefer brooms – slower, but safer.’

As we saw in Deathly Hallows splinching even a small portion of flesh results in person being in grave danger. Then how come the couple was able to survive till the accidental magic reversal squad rescued them?

There are spells that can put Splinched wizards back together.

Splinched wizards can be put back together with spells. When Ron ends up Splinched, he’s only in danger because Hermione doesn’t know if she’s capable of doing the necessary spells correctly. Hermione knows the spells to fix him exist, but doesn’t trust her own ability to do them properly, and is worried she’ll mess them up and make him worse off than if she didn’t.

“It’s all I feel safe doing,’ said Hermione shakily. ‘There are spells that would put him completely right, but I daren’t try in case I do them wrong and cause more damage … he’s lost so much blood already …”
_{^{- Harry Potter and the Deathly Hallows, Chapter 14 (The Thief)}}

One effective way how a Splinched person is repaired is shown. When Susan Bones had Splinched herself in Apparition class, the Heads of House are able to reattach her leg.

“There was a horrible screech of pain and everybody looked around, terrified, to see Susan Bones of Hufflepuff wobbling in her hoop with her left leg still standing five feet away where she had started.

The Heads of House converged on her; there was a great bang and a puff of purple smoke, which cleared to reveal Susan sobbing, reunited with her leg but looking horrified.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 18 (Birthday Surprises)}}

Though they’re not shown being fixed, other students Splinch themselves, and it’s not said that they’re seriously injured or that any deaths occurred - something that’d likely be mentioned.

“Three lessons on, Apparition was proving as difficult as ever, though a few more people had managed to Splinch themselves.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 18 (Birthday Surprises)}}

In most if not all cases of Splinching, the spells can reattach the Splinched-off parts.

Splinching also doesn’t always seem to cause profuse bleeding.

Also, Splinching doesn’t seem to always cause profuse bleeding, like it did in Ron’s case. Before Ron had Splinched a chunk of his arm off, Harry pictured Splinching as comical, despite actually having seen at least one case of it for himself in Apparition class the year before.

“What’s happened to him?’

‘Splinched,’ said Hermione, her fingers already busy at Ron’s sleeve, where the blood was wettest and darkest.

Harry watched, horrified, as she tore open Ron’s shirt. He had always thought of Splinching as something comical, but this … his insides crawled unpleasantly as Hermione laid bare Ron’s upper arm, where a great chunk of flesh was missing, scooped cleanly away as though by a knife.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 14 (The Thief)}}

If Susan Bones was bleeding a lot, it seems likely that would have been mentioned at the time, and even if it wasn’t Harry would likely remember it and not continue picturing Splinching as comical. What the difference between incidents that cause ‘comical’ Splinching and ‘bloody’ Splinching isn’t quite clear. It’s possible, though, that completely Splinched-off body parts are ‘sealed off’ somehow, whereas in Ron’s case he Splinched off a chunk of flesh rather than a complete body part. Not all cases result in drastic hemorrhaging. When returning to the Horcrux search, Ron ends up Splinched again, but this time he’s missing two fingernails and isn’t mentioned as bleeding.

“I didn’t do it so well, Splinched myself again –’ Ron held up his right hand to show two missing fingernails; Hermione raised her eyebrows coldly ‘– and I came out miles from where you were.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 14 (The Thief)}}

Ron doesn’t seem to need medical attention after this, and it’s unclear if he was actually fixed after it. Though Hermione is furious at him and doesn’t really care, Harry still does, but doesn’t seem particularly concerned. It’s entirely possible Ron goes without his fingernails for a while with no ill effects, as there’s nothing about him bleeding, being in danger, or having them fixed.

Possibly Side-Along Apparition Splinching causes more damage.

Another thing to consider is that when Ron got Splinched and was bleeding badly, he hadn’t Apparated himself - Harry tried to transport both him and Hermione by Side-Along Apparition.

“LET’S GO!’ Harry yelled. He seized Hermione by the hand and Ron by the arm and turned on the spot.

Darkness engulfed them along with the sensation of compressing bands, but something was wrong … Hermione’s hand seemed to be sliding out of his grip …

He wondered whether he was going to suffocate, he could not breathe or see and the only solid things in the world were Ron’s arm and Hermione’s fingers, which were slowly slipping away …”
_{^{- Harry Potter and the Deathly Hallows, Chapter 13 (The Muggle-born Registration Commission)}}

All the other instances of Splinching we see are of people who were trying to Apparate themselves. This is the only shown case of Splinching during Side-Along Apparition - it’s possible that being Splinched by someone else trying to Apparate two people can cause more damage in Splinching than one wizard trying to Apparate themselves only and ending up Splinched.

To be realistic, losing even just an arm or a leg would open a major artery, and they would bleed to death within an few minutes. Being literally in half would have them dead very soon.

The fact that the open artery does bleed and is not magically sealed is confirmed when Hermione talks about Ron:

"There are spells that would put him completely right, but I daren’t try in case I do them wrong and cause more damage … he’s lost so much blood already …" (DH)

From the same quote we also know that spells to fix splinching do exist, and the Accidental Magic Reversal Squad will know these spells.

There might be constant monitoring (surveillance) for splinching, but we can rule that out as also from the above situation where Ron is splinched, we know that the ministry didn't detect it, because they wanted to capture them if the could find them.

So the alternative is that there is the equivalent of an emergence phone number, some kind of spell to call for help, and the couple from GOF grew up in the wizarding world and knew how to call for help in time before they died from blood loss.

@QuestionAuthority Medically no, it’s not always the case that losing a limb would cause you to bleed to death quickly - sometimes there is minimal bleeding as the vessels spasm and contract, and muscles in the area tighten. Partial amputations and large wounds are usually much more dangerous with respect to blood loss.