How to determine project complete date?

It really depends on the nature of the backlog/work remaining and the methodology you are using which is not clear on your question.

If you are using SCRUM, you will have to know 2 things.

1. Story point estimated(story point /= story point estimate) entire backlog
2. Team velocity given team is performant and team composition does not change.

(Estimated backlog / Team Velocity) * Duration of Sprints + Buffer - Planned Events/leaves should give you a rough number. Remember schedules in agile are educated guesses not actuals.

If its is waterfall, you need to again have 2 things.

1. Estimated remaining work
2. Resource assignments.

Based on resource assignments and their capacity you will have to calculate the deadlines which are in most cases fixed.

If the methodology is something like XP, you will have to account release plannings and number of iterations into the calculation.

How frequently will you do releases? One thing you could try is to group tickets into tentative releases (or iterations), convert your estimates to points and then estimate a release velocity / iteration velocity.

Prioritisation and minimising dependencies will also be important, especially if your team are not full time. You tagged your question as Scrum. If you want to move towards Scrum then start with product owners, share a backlog with them and define some priorities. It's a mistake to focus on the "project complete" date because all the important things should generally happen early in the programme of work, never at the end.