Unfortunately you are absolutely right here
We have to rely on India's winning because Pakistan is now scheduled to play against Afghanistan and Bangladesh. Both the games are must-win for Pakistan as in case England lose even one of its two remaining games against India and New Zealand respectively. Pakistan will qualify with 11 points.
The second scenario is a bit tricky and interesting, if Pakistan loses one of its remaining games against Afghanistan or Bangladesh, even then Pakistan can qualify but for it to happen England have to lose all its two games.
Pakistan and Bangladesh share seven points on the table. Bangladesh is above because of the Net run rate. Pakistan and Bangladesh now have to play two games. Bangladesh has to play against Pakistan and India. In the case of England’s defeat in coming matches, Bangladesh will qualify for semis if it wins all its remaining two matches.
Sri Lanka will have to play South Africa, West Indies, and India. If Sri Lanka wins all its three games, they will be a point above Pakistan and Bangladesh. And in case England wins all its three games, it will be sharing 12 points with Sri Lanka. Team with the better run rate will qualify.
Pakistan has to defeat Bangladesh at any cost.