There was a study in 2009 that showed that Early Risers Crash Faster Than People Who Stay Up Late. From the article:

Researchers Christina Schmidt and Philippe Peigneux and their colleagues first asked 16 extreme early risers and 15 extreme night owls to spend a week following their natural sleep schedule. Then subjects spent two nights in a sleep lab, where they again followed their preferred sleep patterns and underwent cognitive testing twice daily while in a functional MRI scanner.

An hour and a half after waking, early birds and night owls were equally alert and showed no difference in attention-related brain activity. But after being awake for 10 and a half hours, night owls had grown more alert.

Another article on the same research: Night Owls Are More Productive than Early Birds.

The original research published in Science was: Homeostatic Sleep Pressure and Responses to Sustained Attention in the Suprachiasmatic Area (requires creating a free login to read the full report).

This is what we call Chronotype, Clodoré et al. has also done research about alertness differences.

Chronotype is an attribute of animals, including human beings, reflecting at what time of the day their physical functions (hormone level, body temperature, cognitive faculties, eating and sleeping) are active, change or reach a certain level.

[..]

Clodoré et al., France, 1986, found differences in alertness between morning and evening types after a two hour sleep reduction.

— Wikipedia - Chronotype

Further outlined in an abstract we find:

Morning-types (MT) and evening-types (ET) differed only during the morning: ET fell asleep more frequently at 1000 hr and 1200 hr and rated lower self-alertness on arising than did MT.

Twelve subjects were given the protocol of a 2-hr sleep reduction (both in delayed bedtime and advanced rising time conditions). At 0700 hr, MT rated their alertness lower when they had only just gotten up (delayed bedtime condition) than when they had been awake for 2 hr (advanced rising time condition).

In contrast, ET had the same low level of alertness at 0800 hr, independent of the time elapsed since arising. On average the advanced rising time condition affected the general pattern of alertness more than did delayed bedtime.

Clodoré M, Foret J, Benoit O. - Diurnal variation in subjective and objective measures of sleepiness: the effects of sleep reduction and circadian type

We can also look at circadian rhythm sleep disorders; if we look at Delayed Sleep Phase Syndrome we see:

People with DSPS can be called extreme night owls. They feel most alert and say they function best and are most creative in the evening and at night. DSPS patients cannot simply force themselves to sleep early. They may toss and turn for hours in bed, and sometimes not sleep at all, before reporting to work or school. Less extreme and more flexible night owls, and indeed morning larks, are within the normal chronotype spectrum.

— Wikipedia - Delayed Sleep Phase Syndrome

The Advanced Sleep Phase Syndrome is less common, thus it's alertness hasn't been mentioned yet.

From http://archpedi.jamanetwork.com/article.aspx?articleid=379301

The preference for particular sleep timing is called “chronotype”; some people are morning types and choose to go to bed early and wake up early, whereas others are evening types and go to bed late and awaken late.

As Tom points out, Chronotype is the key word here - Wikipedia has a slab of text on what the research says... examples include (choosing those that are not in other answers):

• Horne and Östberg, 1976, found that morning types had a higher daytime temperature with an earlier peak time than evening types and
  that they went to sleep and awoke earlier, while no differences in
  sleep lengths were found. They also note that age should be
  considered in assessments of morningness and eveningness, noting how
  a "bed time of 23:30 may be indicative of a Morning type within a
  student population, but might be more related to an Evening type in
  the 40–60 years age group" (Horne & Östberg, 1976, p. 109).
• Duffy et al., US, 1999,[11] investigated "changes in the phase relationship between endogenous circadian rhythms and the sleep-wake
  cycle," and found that while evening types woke at a later clock hour than morning types, morning types woke at a later circadian phase.
  Baehr et al., US, 2000,[12] found that, in young adults, the daily
  body temperature minimum occurred at about 4 a.m. for morning types
  but at about 6 a.m. for evening types. This minimum occurred at
  approximately the middle of the eight hour sleep period for morning
  types, but closer to waking in evening types. Evening types had a
  lower nocturnal temperature. The temperature minimum occurred about a half hour earlier in women than in men.

But as for the key point about if it's nature or nuture...