Dart

The secret ingredient is a big cache.

import 'dart:math';
import 'package:collection/collection.dart';
import 'package:more/more.dart';

Map<String, Point<num>> mapper(List<String> list) => {
      for (var r in list.indexed())
        for (var c in r.value.split('').indexed())
          c.value: Point<num>(c.index, r.index)
    };
var dirMap = mapper(['_^A', '<v>']);
var numMap = mapper(['789', '456', '123', '_0A']);

var lenCache = <(String, int), int>{};
int len(String code, int level, isNum) =>
    lenCache.putIfAbsent((code, level), () => _len(code, level, isNum));
int n(p, isNum, level) => len(getMoves(p[0], p[1], isNum), level - 1, false);
int _len(String code, int level, isNum) => (level == 0)
    ? code.length
    : 'A$code'.split('').window(2).map((p) => n(p, isNum, level)).sum;

String getMoves(String f, String t, bool isNum) {
  var map = isNum ? numMap : dirMap;
  var (from, to) = (map[f]!, map[t]!);
  var mv = to - from;
  var s = <String>{};
  var x = ''.padRight(mv.x.abs() as int, mv.x.sign == 1 ? '>' : '<');
  var y = ''.padRight(mv.y.abs() as int, mv.y.sign == 1 ? 'v' : '^');
  // avoid '_', dislike '<'
  if (Point(from.x, to.y) != map['_']!) s.add('$y${x}A');
  if (Point(to.x, from.y) != map['_']!) s.add('$x${y}A');
  return (s.length < 2 || mv.x > 0) ? s.first : s.last;
}

int solve(String code, int level) =>
    len(code, level + 1, true) * int.parse(code.skipLast(1));

part1(List<String> lines) => lines.map((e) => solve(e, 2)).sum;
part2(List<String> lines) => lines.map((e) => solve(e, 25)).sum;