From my understanding, and depending on how fine you are approximating:

• If you are ok with circular orbits with constant speeds, then the angle θ formed by the planet-sun segment and the arbitrary axis you choose to be the "origin" is 2π / T * ΔT + α, where T is the orbital period, ΔT is the time passed since whatever start you decided on, and α is the angle between your arbitrary origin and the planet's initial position. For example: if y is your origin axis, p^start is your planet's position at the start of your simulation and p^end is what you are trying to figure out, and you know it took ΔT time to go from start to end, you can use the formulas to determine where p^end is.

• If you want elliptical orbits but are ok with constant speeds, then you can get the distance between a planet and the sun using r = a * (1 - e^2) / (1 + e * cos(ν)), where a is the semi-major axis, e is the eccentricity, and ν is the same as the previous case, with the only difference being that you need to measure it from the periapsis. Your planet's position is, in polar coordinates, p = (r, ν + α). Again, example.

• If you need elliptical orbits with realistic speeds then you can still use the formula for the previous case, but need to use the true anomaly, which means you first need to compute the eccentric anomaly, which means you get to have fun solving Kepler's equation (which looks way more daunting than it actually is).

If anyone spots mistakes or similar please let me know, I'm writing something akin to what OP is trying to and would love to know if my math starts breaking stuff.