Partition inside an open probate is something I want to answer carefully, because the procedural specifics vary enough by state that I would confirm anything here with a probate attorney before acting on it. That said, here is how it generally works.
A partition action is typically a separate civil filing, not something that runs through probate court. The probate case handles the estate's administration, and the partition case handles the dispute over what happens to a specific asset inside it. Some states allow a probate judge to handle both if the property is still part of an open estate, but most route the partition to a civil court. That means two simultaneous proceedings, which adds time and legal cost.
How courts resolve it: the holdout heir cannot simply block a sale indefinitely. A partition action asks the court to either physically divide the property (almost never possible with a single-family house) or order a partition by sale, where the court forces a sale and distributes proceeds according to ownership shares. The sibling who wants to keep it can bid at that sale, but if they cannot finance a buyout privately, they are unlikely to outbid a third-party buyer at a court-ordered sale either.
Timeline in practice varies widely. A straightforward uncontested partition might resolve in a few months. A contested one with a motivated holdout can run a year or more.
A buyer under contract at this point generally has no standing to compel the partition. The contract is with the estate or the selling heirs, and the buyer's leverage is the contract itself. Whether that contract survives a long delay depends on how it was written, specifically whether it contains a timeline contingency tied to clear title.
A probate or real estate attorney in the relevant state is the right person to confirm which court takes jurisdiction and what the realistic timeline looks like in that county.
What state is the property in? That changes the answer more than anything else here.