Laws relevant to when the 10th of Teves falls on Erev Shabbos:
It is possible for the 10th of Teves to fall on Erev Shabbos. It is the only fast in the Jewish calendar that can fall on Erev Shabbos. When the 10th of Teves falls on Erev Shabbos, the fast is not delayed to another day.
Haircuts: Even those who are accustomed not to get a haircut on the day of the fast may do so on a fast day that falls on Erev Shabbos.
Until what time must one fast: When the 10th of Teves falls on Erev Shabbos, one fasts until after nightfall of Shabbos, as is the law regarding all fasts that one must fast until after Tzeis Hakochavim. This applies even if one Davens Maariv and accepts Shabbos early prior to nightfall, nonetheless he must continue fasting until nightfall arrives. [There is no obligation to make Kiddush and begin eating immediately after nightfall and one may delay as necessary, such as to Daven Maariv and walk home from Shul, or wait for guests to come, or to delay past the seventh hour, as will be explained. Nonetheless, initially it is proper for one to arrange for Kiddush to be made as soon as possible after nightfall in order not to elongate one’s state of oppression on Shabbos, and thus one may Daven Maariv early even prior to nightfall for this purpose, as will be explained.]
Mincha Erev Shabbos: Prior to Mincha, Hodu and Patach Eliyahu are recited as usual. The Torah is read in Mincha, as usually done on a fast day. However, Tachanun [and Avinu Malkeinu] is not recited by Mincha. Aneinu is recited in Mincha. One is to be careful to Daven Mincha, including Chazaras Hashatz, before sunset, in order not to enter a question of reading the Torah and saying Aneinu after sunset. Nonetheless, in a time of need the Torah may be read, and Aneinu may be said, up until nightfall.
When to Daven Maariv: One may begin Maariv after sunset, prior to Tzeis Hakochavim, in order to conclude Maariv and be able to make Kiddush and eat at the earliest possible time. In such a case, one is to make sure to read all three paragraphs of Kerias Shema for the sake of the Mitzvah again after nightfall, prior to starting the meal. Certainly, one may Daven Kabalas Shabbos prior to nightfall, and so is encouraged to be done by all Shuls in order not to delay the fast unnecessarily.
Kiddush Between 6:00-7:00 [or 5:40-6:40 in Israel]: Although it is generally accustomed amongst many Chassidim to avoid making Kiddush between six and seven, nevertheless in years that the fast of the 10th of Teves falls on Shabbos many are accustomed to being lenient in this and make Kiddush between six and seven if necessary. Others however relate that the custom is to be stringent not to make Kiddush between six and seven even in such a scenario that will cause the fast to be delayed. Practically, relevant to one’s geographical location, Shuls and individuals should initially schedule their Davening of Maariv and coming home from Shul in a way that it will allow them to make Kiddush prior to the 6th hour. In a time of need, however one may be lenient to make kiddush after the sixth hour.
Summary of options for when to break the fast:
One is obligated to fast until nightfall. The following are the options available in how to navigate breaking the fast with Kiddush as soon as possible after nightfall:
May one choose to make Kiddush and eat a snack after nightfall prior to Davening Maariv?
Background: There is a dispute mentioned regarding one who has finished Davening Maariv of Friday night prior to nightfall if he must continue fasting until nightfall. The first opinion rules that one must still continue fasting until nightfall [as all public fasts must be complete fasts]. The second opinion rules that it is forbidden to continue fasting past the conclusion of Maariv, even if before nightfall, as at that time one has already fully accepted Shabbos, and it is forbidden to fast on Shabbos, with intent of fasting, for even one moment. Now, although every public fast must be completed for the entire day [until nightfall], this second opinion holds that after Maariv is already considered a new day and thus there is no need to wait until nightfall out of obligation to complete the fast. [Admur ibid] Regarding a public fast Admur rules like the first opinion being that it is obligatory for one to fast until nightfall by a public fast. So, rules also Rama and Mishneh Berura 249. Regarding a private fast Admur rules like the second opinion.
Other opinions: Some Poskim rule the Torah is not read by Mincha in order so people have time to prepare for Shabbos. [Beis Yosef 550, brought ion M”A ibid]
Regarding the prohibition to Daven both Mincha and Maariv on a single day within the period of Plag Hamincha: See Admur 267:2; M”A 267:1; M”B 267:3; Biur Halacha 267:2 “Umiplag”; Piskeiy Teshuvos 267:2 footnote 19. Some Poskim however are lenient to allow a congregation to Daven both prayers, one after the other, within the period of Plag Hamincha, if it will be difficult to gather the congregation again afterwards for Maariv. [Regarding weekdays: M”A 233:6; M”B 233:11; Poskim in Kaf Hachaim 233:12; Regarding Erev Shabbos: Derech Hachaim Arvis 1, brought in M”B 267:3; Biur Halacha ibid; Shraga Hameir 5:21; Kinyan Torah 4:27; See Piskeiy Teshuvos 233:3; 267:2] Other Poskim however rule one may only be lenient to do so if one will Daven Maariv after sunset. [M”B 267:3; Biur Halacha ibid regarding Erev Shabbos; Vetzaruch Iyun from M”B 233:11 who records the ruling of the M”A without argument.] There are communities that follow the former opinion and Daven both Mincha and Maariv within Plag Hamincha. [See Shraga Hameir 5:21; Kinyan Torah 4:27]
The law by a fast day and by Purim: The above ruling of the Poskim refers to a regular day and not a fast day. However, on a fast day, one can possibly argue that it would be prohibited to even taste any food prior to Maariv unless it is a time of great need, just as rule the M”A 692:7 himself and other Poskim rule regarding tasting food prior to hearing the Megillah, that one may not do so unless it is a time of great need. The reason recorded behind this stringency is unclear. [The M”A ibid states that it is because we find that even prior to the night Shema some authorities rule that one may not even snack, lest he fall asleep. Furthermore, Megillah is not similar to other Mitzvos by which we allow snacking beforehand, as all Mitzvos are pushed off in face of Megillah reading. See Levush 692:4; Nitei Gavriel 32 footnote 1 that the reason the eating law before Megillah is more severe than before other Mitzvos is because they suspected one may come to continue in eating due to the hunger caused by the fast.] Now, while it is possible to understand that the reason behind his stringency is due to the fact that it is a fast day, in which case we then prohibit even tasting, it is also possible to argue that the stringency is due to the greatness of the mitzvah of reading the Megillah. Practically, I have not found any discussion on the subject and therefore conclude as does the M”A ibid himself regarding Taanis Esther, that he should not even taste food unless it is a time of great need, although it is strongly possible to argue that even on a fast day tasting is permitted even initially.