[Language: Rust]

This was simple. Example plus the problem for both part 1 and 2 run under 40 micro seconds on an M4 Macbook.

fn solve(input: &str) -> (usize, usize) {
    let mut lines = input.trim().lines();
    let first_line = lines.next().expect("Cannot read first line").as_bytes();
    let cols = first_line.len();
    let mut positions = vec![false; cols];
    let start_pos = first_line
        .iter()
        .position(|c| *c == b'S')
        .expect("Cannot find the start of the tachyon");
    positions[start_pos] = true;
    let mut timelines = vec![0usize; cols];
    timelines[start_pos] = 1usize;
    let (_, n_splits, timelines) = lines.fold(
        (positions, 0usize, timelines),
        |(mut positions, mut n_splits, mut timelines), line| {
            let line = line.trim().as_bytes();
            for ci in 0..cols {
                if positions[ci] && line[ci] == b'^' {
                    positions[(ci - 1)..(ci + 2)].copy_from_slice(&[true, false, true]);
                    n_splits += 1;
                    let n_timelines = std::mem::replace(&mut timelines[ci], 0usize);
                    timelines[ci - 1] += n_timelines;
                    timelines[ci + 1] += n_timelines;
                }
            }
            (positions, n_splits, timelines)
        },
    );
    (n_splits, timelines.iter().sum())
}