The Hanafi school of thought has the view that an adult woman can carry out her own marriage, without the consent or involvement of a wali.
They base it on the following verses of the Quran, arguing that the woman is given the right to contract her own marriage and the consent of the wali is not mentioned as conditional. (See Ahkam ul Quran by Al Jasas for details):
Quran 2:230 And if he has divorced her [for the third time], then
she is not lawful to him afterward until [after] she marries a husband
other than him. And if the latter husband divorces her [or dies],
there is no blame upon the woman and her former husband for
returning to each other if they think that they can keep [within] the
limits of Allah . These are the limits of Allah, which He makes
clear to a people who know.
Quran 2:234 And those who are taken in death among you and leave
wives behind - they, [the wives, shall] wait four months and ten
[days]. And when they have fulfilled their term, then there is no
blame upon you for what they do with themselves in an acceptable
manner. And Allah is [fully] Acquainted with what you do.
They also interpret the following hadith to support their view:
A woman without a husband has more right to her person than her
guardian, and a virgin's consent must be asked from her, and her
silence implies her consent. [Saheeh Muslim]
A woman came to Allah's Messenger (ﷺ) and said, "O Allah's Messenger
(ﷺ)! I have come to give you myself in marriage (without Mahr)."
Allah's Messenger (ﷺ) looked at her. He looked at her carefully and
fixed his glance on her and then lowered his head. When the lady saw
that he did not say anything, she sat down. A man from his companions
got up and said, "O Allah's Messenger (ﷺ)! If you are not in need of
her, then marry her to me." The Prophet (ﷺ) said, "Have you got
anything to offer?" The man said, "No, by Allah, O Allah's Messenger
(ﷺ)!" The Prophet (ﷺ) said (to him), "Go to your family and see if you
have something." The man went and returned, saying, "No, by Allah, I
have not found anything." Allah's Apostle said, "(Go again) and look
for something, even if it is an iron ring." He went again and
returned, saying, "No, by Allah, O Allah's Messenger (ﷺ)! I could not
find even an iron ring, but this is my Izar (waist sheet)." He had no
rida. He added, "I give half of it to her." Allah's Messenger (ﷺ)
said, "What will she do with your Izar? If you wear it, she will be
naked, and if she wears it, you will be naked." So that man sat down
for a long while and then got up (to depart). When Allah's Messenger
(ﷺ) saw him going, he ordered that he be called back. When he came,
the Prophet (ﷺ) said, "How much of the Qur'an do you know?" He said,
"I know such Sura and such Sura," counting them. The Prophet (ﷺ) said,
"Do you know them by heart?" He replied, "Yes." The Prophet (ﷺ) said,
"Go, I marry her to you for that much of the Qur'an which you have."
The only condition they attach is that the husband should be a suitable\equal match to the wife, and in case of an unequal match the awliya can dispute the marriage and a Qadi can be persuaded to annul it. In your case the parents themselves agree to the marriage, just not the timing.
From the Hanafi manual Hidayah:
The nikah of a sane and major freewoman stands concluded, when it is
with her consent, even if the wali (guardian with legal authority
granted by the shareah) did not undertake this contract. This is so
according to Abu Hanifah and Abu Yusuf (God bless them) recorded as
the Zahir al- Riwayah. It is narrated from Abu Yusuf (God bless him)
that it is not concluded, while according to Muhammad (God bless him),
it is concluded but is suspended (mawqiif, subject to ratification by
Malik and al-Shafil (God bless them) said that nikah is not concluded
at all through a statement of women, because a nikah is intended to
meet certain objectives and delegating such authority to them upsets
these objectives. Muhammad (God bless him) said that such upsetting
basis objectives is remedied after ratification of the contract by the
wali. The basis for permissibility (according to the Zahir al-Riwayah)
is that she has undertaken an act that pertains to something that is
purely her personal right, and she possesses the legal capacity to do
so being sane and in possession of discretion. It is for the same
reason that she can undertake transactions in wealth and possesses the
right to choose a husband. The wali is asked to undertake her marriage
so that she is not characterised as being immodest.
Thereafter, according to the Zahir al-Riwayah, ther is no difference
between a husband who is equal in status to her and one who is not
equal in status. It is also reported from Abu Hanifa and Abu Yusuf
that in the case of a husband of lesser status it is not permitted,
because there are many matters (between husband and wife) that cannot
be resolved by resort to law. It is reported that Muhammad withdrew
his opinion and upheld the one followed by the two jurists.
If you go with the Hanafi view you can arrange your own marriage with witnesses etc. and register it with the court.
The Shafi school of thought does not consider a marriage without a wali to be valid. They base it on the following verse, arguing that the reason the Quran admonishes the walis to not prevent women is because they have the right:
Quran 2:232 And when you divorce women and they have fulfilled
their term, do not prevent them from remarrying their [former]
husbands if they agree among themselves on an acceptable basis. That
is instructed to whoever of you believes in Allah and the Last Day.
That is better for you and purer, and Allah knows and you know not.
They also base it on the following hadith:
The Messenger of Allah said: “Any woman whose marriage is not
arranged by her guardian, her marriage is invalid, her marriage is
invalid, her marriage is invalid. If (the man) has had intercourse
with her, then the Mahr belongs to her in return for his intimacy with
her. And if there is any dispute then the ruler is the guardian of the
one who does not have a guardian.” Sunan Abu Daud, Sunan Ibn
Majah, Jami Tirmizi
The Shafis also use the above hadith to derive that if a Wali refuses to marry the woman, then the state can resolve the dispute and become her wali. In a non-muslim country, a non-muslim judge can not act as your wali, but you can approach an Imam of a mosque or a pious and knowledgeable Muslim to act instead.
From the Shafi manual, Umdat as Salik:
Whenever a free woman asks to marry a suitor who is a suitable match
(def: m4) (0: by telling her guardian, "Marry me to him"), the
guardian must marry her to him (0: whether she is a virgin or
nonvirgin, and whether prepubescent or not). The Islamic magistrate
(A: i.e. judge) marries her to such a groom if the guardian:
(1) in the presence of the magistrate refuses to marry her to the
(2) is on a journey farther than 81 km/5O mi. from home;
(3) or is in a state of pilgrim sanctity (ihram) (0: for hajj, 'umra,
or both) (dis: j3.20).
In such eases, the guardianship does not devolve to the next most
eligible in the m3.7 order of lawful guardians. If (non-(2) above) the
guardian is on a journey of less than 81 km./SO mi. from home, the
bride may not be married to someone without the guardian's leave.