Smart move.
First, if the environmental change is gradual, there is no need to consider such a drastic reduction in population. It is true that in each generation there would be fewer of the non-modified population, but at the same time there would be more of the modified population. So the total population need not decrease at all.
The bottleneck scenario you are describing would not likely occur as a result of gradual environmental change, but as the result of a single catastrophe. This is the sort of thing one sees when a new pesticide or antibiotic is used against a population that has never been exposed to it before. There is an immediate crash in population size with only a few that happen to have more resistance surviving.
The second scenario is possible and gives rise to what is called an adaptive radiation: several different species all arising from the same ancestor within a short period of time. These may have been more common than we know because we can only work with surviving lineages and those we discover in the fossil record. And these represent only a small proportion of all the species that have ever emerged.
For the first scenario, it is not necessary that all the mutations be the same. What you need is for all the different mutations to accumulate in some individuals. If each one is favorable, each will spread through the population via reproduction. So at some point what will occur is that Johnny will inherit favorable mutation #1 from mummy and favorable mutation #2 from daddy. And in another family Jenny will inherit favorable mutation #3 from mummy and favorable mutation #4 from daddy.
Then it is possible for some of Johnny and Jenny's kids to inherit all four favorable mutations.