From a re-read of the chapters; there is no evidence that the Ministry was not alerted by the Trace.
Who's There to Care?
Remember that almost everyone who is anyone is at the Triwizard Tournament. This includes:
- Prime Minister Fudge,
- Albus Dumbledore,
- Dignitaries from abroad,
- Misc. members of the Ministry (Bagman, Kingsley? etc.)
Time is of the Essence
Harry was gone, lets say about 30-45 minutes in total1. This includes,
- Cedric's death
- Flesh, Blood and Bone Ceremony
- DEs appearing
- Voldemort's convenient recount of his life(?) for the past 12 years
- Harry and Voldemort's duel
- Priori Incantatum
It is safe to assume that Cedric's body was still warm when Harry got back. This doesn't leave much time for the Ministry to do much about it.
By the time Harry had escaped, it would not have taken long for the DEs and Voldemort to 'leg-it'.
The Trace is not 'All-knowing'
From what we've seen and know about the Trace is that it knows two things for sure:
1. Magic has been used by, or around, an underage wizard.
2. The (approximate) location of the incident.
That's it. It doesn't know for sure sure who the underage wizard was2, or the nature of the magic (very powerful dark magic or a simple Lumos).
So in effect, even if the Ministry was alerted, what they know for sure is that somewhere near Little Hangleton Cemetery some magic was used by, or near, an underage wizard.
The spectators of the Triwizard tournament already know that something is wrong, and that Harry and Cedric appear to be missing, and the extra information from the Ministry would have been inconsequential at that point...
There is no way to know whether or not the Ministry was alerted, and even if they were, it would not have brought much help at that time3.
1. Complete guesstimate from me!
2. In the case of Harry VS Birthday cake and Harry VS Dementors; they know that the magic was used in Harry's home town, so it's not difficult to figure out who it was.
3. Maybe afterwards to corroborate Harry's story.
4. Some light reading:
- How strict is the Decree for the Reasonable Restriction of Underage Sorcery?
- I don't quite understand the Trace with regards to underage Wizards performing magic