I was Baptized Baptist and we were very particular about it: immersion, one time, backwards. No ifs, ands, or buts. I am extremely terrified of putting my head under water; but, for whatever reason, I didn’t have any problem with getting baptized at all.
I think that between already being nervous about being up front in front of the whole church and the fact that the Pastor knew I was terrified of water and kind of “snuck it up on me” really helped.
I have seen Baptists, especially in Europe, use a more Anabaptist mode: one or three times forward. I also know that in my current church (the C&MA) we are credo-baptist in our position and greatly prefer the Baptist mode for our baptisms, but we will allow different modes where it is practical to do so. I think you’ll find the view to be the same in most other non-Baptist evangelical traditions.
There’s also the fact that, even in most Baptist churches, Baptism is not seen as being strictly necessary for salvation and if someone is genuinely so terrified of water as to render baptism by immersion unsafe (because, let’s face it, someone flailing around in the water, even in a bathtub style baptismal font, is unsafe), then nobody’s going to force it on them.