My advice is that you should model Holacratic behaviors to support the power shift your organization is trying to achieve, but that does not limit your ability to get updates and drive action on the project. Here are some of the tools available to you as lead link:
1. Like any circle member, lean on the duties found in Article 4. In particular, 4.1.1(b) and (c) should be really useful. Ask the person what other projects are prioritized of the one that is not receiving attention. Ask the person when he or she expects to have it done.
2. Also, remember that each project should have a next-action associated with it. Ask what the next-action is on that project. If there is one, you will be able to help evaluate whether that action is going to move the project forward and is truly the next step to take or whether there is some intermediate action that should be included. If there is not one, ask the person to define a next-action. This will reinforce the best practices of Getting Things Done and allow the role-filler to come into compliance with his/her responsibilities as outlined in Article 1. See especially 1.2 . . . does the person have a trusted system (1.2.4) and is he/she defining next-actions (1.2.3) effectively? This step is also available not just to the lead link but to any circle member.
3. As lead link, you are accountable for the priorities of the circle. If you think the neglected project is more important and needs to be elevated in priority, you can say so. This is ideally done in a collaborative way and with an understanding of the other projects and priorities, but ultimately, you can direct the resources (including employee time) toward the priorities that you think are most deserving.
Hope these ideas help get you started. To answer your final question, I don't think you as LL can expect any particular project to move at any particular pace unless you have received some any indication to that effect using the tools above or unless common sense would make it obvious that it should have moved at that pace (i.e. estimated tax payment due on June 15 either makes the deadline or doesn't). However, you always have the authority to ask for more detail, ideally framed as above, and to drive action using your role's authority and your rights as a member of the circle.