It is known that Harry could have used Draco's wand and it have worked for him, but the wand would have not been the best match. This can be seen from the fact that in book 7 Harry used hermione wand and it worked for him, but when he used Draco's it worked better than Hermiones(because he had won it), but it still didn't work as well as his first wand.
"Harry looked down at the hawthorn wand that had once belonged to
Draco Malfoy. He had been surprised, but pleased, to discover that it
worked for him at least as well as Hermione's had done." —Harry
Remember his wand is special since only 2 like it were made.
"The wand chooses the wizard... it's not always clear why."
—Ollivander talking to Harry Potter
We can't really say why the wand chooses the wizard as said by ollivander in that qoute. This qoute came from the Harry Potter wikia.
Also his wand was special, since he was Harry Potter the boy who lived his wand going to probably be related to Voldemort's wand. Since Draco's wasn't related it isn't likely that Harry would have gotten it.
His connection to his wand can be seen from this qoute from the Harry Potter wikia.
He had tried out many other wands first with little success, but felt
a warm feeling upon being handed the holly wand. Despite his wand
being brothers with Lord Voldemort's wand, noted by Ollivander after
Harry's wand had been chosen, Harry generally did not put much thought
into this connection at first, regarding the relationship as something
that the wand could not help, any more than he could help being
related to his aunt. http://harrypotter.wikia.com/wiki/Harry_Potter's_wand
So all in all as long as he was Harry Potter there would have been no change in wand since his wand was given to him because of the connection Harry and Voldemort shared. Remeber the shared cores of Fawkes feathers.
These qoutes from the wikia should help you see that he would never got Draco's wand.
"Wands are only as powerful as the wizards who use them...." —Hermione
explaining to Harry Potter[src]
...... In almost all documented cases the
wand chooses the wizard who matches its character because the wizard
may have difficulty or be unable to perform magic if his or her wand's
characters conflict, or the magic may be sub-par to magic performed
with a matching wand. Although Ollivander says that wizardkind can
channel their powers through almost anything, optimum results can only
be achieved with those they have a natural affinity for. When trying
out different wands, a wand that is not suited to its wielder will
usually not do anything at all. However, a warm feeling is an
indication of having chosen the right wand, and the wand sometimes
emits a few sparks or does some similar, small sign of
Wands can take new masters. When a wand's master is disarmed, stunned,
or killed, the wand may accept the old master's attacker as its new
master, swaying its allegiance over to the wizard who won it from its
preceding owner. This phenomena is mostly observed in the Elder Wand,
which, according to Albus Dumbledore, only respects power. The Elder
Wand will transfer its allegiance even if its defeated master is not
in possession of it at the time. No other wand is known to share this
trait. Most wands are attached to their masters and will not easily
discontinue their allegiance with the previous master, if at all.
Indeed the original bond seems unlikely to be severed; if it were,
then a defeated witch or wizard would be better served to acquire a
new wand rather than continue using it because it would no longer
perform at its peak for them--yet they don't. Furthermore, wands
forcibly taken in circumstances of purely competitive or friendly
nature or stolen without attacking the previous master will not change
We can see that Draco's wand did pick him and only him in the qoute from the Harry Potter wikia and book 7.
Draco: "You have something of mine. I'd like it back." Harry: "What's
wrong with the one you have?" Draco: "It's my mother's. It's powerful
but, it's not the same... doesn't quite, understand me. Know what I
mean?" — Draco asking Harry for his wand
Also the size of his wand might make a difference. This is from the wikia, so you might say its conjecture, but the wikia is somewhat(usually) accurate.
Draco Malfoy's wand - Hawthorn, unicorn hair, 10 inches.
was 11 inches long, made of holly and had a phoenix feather core Because numbers have known magical properties, a wand's length, much
like its core, may symbolize something its owner. For example, the
wand of Tom Riddle, a reputed Dark wizard, measures 13½ inches, and
thirteen is commonly an omen of misfortune or evil in the folklore and
religion of various cultures around the world. Wands that are
unusually short, anything less than 9", tend to belong to people
"lacking" in personalities, Dolores Umbridge being a known
This information gotten the wikia seems to prove that Harry could have used Draco's wand. But it would never have picked him.