The Rambam writes in Hil. Eruchin (4:24) that when one is makdish the field and fails to redeem it, it goes to the kohanim by yovel. The Rambam writes - ואם היה ראש השנה של יובל בשבת שהרי משמר יוצא ומשמר נכנס,תנתן למשמר היוצא. The Rambam clearly holds that Yovel begins on Rosh Hashana so that the kohanim serving on Rosh Hashana have the rights to the field. Both the Lechem Mishneh raises the point that the Rambam in Hilchos Shemitin V'Yovel writes that Yovel begins on Yom Kippur, not Rosh Hashana, so why does he make it dependent on the kohanim of the mishmar of Rosh Hashana?
In truth, there is a machlokes tanaim about this issue. The first mishna in Rosh Hashana says that the first day of Tishrei is Rosh Hashana for Yovel. The gemara 8b asks that yovel doesn't begin on Rosh Hashana, it begins on Yom Kippur? The gemara responds that our mishna which says that Yovel begins on Rosh Hashana follows the opinion of R. Yochanan Ben Broka who says that in the Yovel year from Rosh Hashana until Y.K. the slaves no longer work, but also may not go home, rather they eat and party until the sound of the shofar on Y.K. at which point they return home. Both Rabbeinu Chananel and Tosafos explain that the gemara makes our mishna that considers R.H. to be the start of yovel to follow R. Yochanan Ben Broka, but the Rabbonon disagree and hold that yovel only begins on Yom Kippur.
Going back to the Rambam, the Mishnah L'melech points out that the question of the Lechem Mishna doesn't make sense. The Lechem Mishna understood that the concept of מר"ה עד יוה"כ לא היו עבדים נפטרין לבתיהן ולא משתעבדין לאדוניהם אלא אוכלין ושותין ושמחין וכו' כיון שהגיע יוה"כ תקעו ב"ד בשופר נפטרו עבדים לבתיהן is the opinion of the Rabbonon who hold that Yovel begins on Y.K. but according to R. Yishmael the avadim would even return home on R.H. Based on that assumption the Lechem Mishna asks that the Rambam who quotes that statement in Hilchos Shemitah contradicts the ruling in Hil. Eruchin of Yovel starting on R.H. But in truth, the Rambam is simply paskening that yovel begins on R.H. like R. Yishmael, yet the avadim don't return home until Y.K.
We still need to understand the concept of Yovel beginning on R.H. but the avadim not being able to return home until Y.K. Why would that be? The Minchas Chincuh (end of mitzvah 332) cites the Turei Even who says that yovel is dependent on blowing shofar so the yovel cannot go into effect without the shofar being blown on Y.K. But, when the shofar is blown on Y.K. the entire year gets kedushas Yovel even retroactively starting for R.H. Therefore the avadim cannot be forced to work because they may retroactively be freed if yovel goes into effect, but they cannot go home since the kedusha is still pending until the shofar is blown on R.H.
The approach of the Turei Even works to explain the Rambam, but it seems to me that the true peshat in the Rambam is simpler and can be understood from what he writes in the pirush hamishna. The Rambam writes - אע"פ שדיני יובל אינן מתחילין אלא מיום העשירי לתשרי לפי שאמר השי"ת "ביום הכפורים תעבירו שופר", אבל מראשית השנה יתחיל חשיבותה ותסתלק העבודה מן העבדים כדבר השי"ת וקדשתם את שנת החמשים שנה. The Rambam does not seem to support the notion of kedushas Yovel going into effect by the blowing of shofar retroactively from R.H. as the Turei Even explains. Rather, the Rambam holds that there are 2 dinim in Yovel. There is the kedusha of yovel and there is the halachos of yovel. The kedusha of yovel goes into effect on R.H. and that entitles the avadim to not have to work, but they are not set free. The dinim of yovel only begin on Y.K. and that is when the avadim go free (According to this approach the fields of hekdesh going to kohanim should be on Y.K. not R.H. as the Rambam writes in Hil. Erchin?)