The Conservative movement requires immersion in a kosher mikvah, circumcision (for male converts), and evaluation by a beit din (a court of three). Immersion and circumcision require at least two witnesses. There might be exceptions to some requirements (for example, medical issues that make circumcision a bad idea), but this is the baseline.
A natural body of water (of sufficient size) can be a kosher mikvah, but do note that immersion is in the nude, so public bodies of water might not be suitable for reasons of modesty and/or public-nudity laws. The laws of mikvah are complicated enough that you shouldn't try to make your own.
Conversions are overseen by a beit din, a court of three, who (I understand) are also the witnesses to the immersion. I didn't find anything that specifically requires them to be rabbis, but as a practical matter, at least in the US they all are -- to be accepted as a Conservative convert you have to work with a Conservative rabbi, and that rabbi assembles the beit din from among his peers.
After immersing (while still in the water, I believe), the convert makes the blessing you quoted (al ha-tevilah is the only name I know for it) and also the shehechiyanu. The witnesses listen.
This list of responsa from the Rabbinical Assembly addresses many questions related to conversion, but has no entries about exceptions for the mikvah, beit din, or witnesses. I realize that arguing from a negative is iffy, but I've known several Conservative (adult) converts over the years and this process is what they've described.
Since you mentioned the example of a child of a Jewish father (and non-Jewish mother), you might be interested in this ruling on child conversions. Note that even in cases where there is room for leniency, they still require the mikvah. There is discussion there about whether a b'racha is said for a minor or whether it is altered. I didn't find a definitive ruling on that.