I think @Paul Goyes's solution only works if you have created a 1-mile polygon buffer layer for your point, similarly to Select by Location.
I don't think the nearest neighbour algorithms are aimed at what you're trying to do either - they generally keep looking to find the closest match(es) and are about comparing what the best match is for each of multiple source points. Not just selecting everything within a fixed distance from a single point.
If you don't want to create another layer per Paul's solution, the easiest way to select everything from one layer given a distance from another layer may be using QGIS expressions in the Select by Expression dialog box.
In this example I have a single green point labeled POINT (layer
point) Now I want to select everything in another layer (
otherpoints) that is within a 20m radius of POINT - shown as a pink circle in this diagram.
I selected the layer
otherpoints and used the following expression in Select Features by Expression:
(Note the layer reference, being
point in this instance, is case-sensitive)
As you can see below this selected the relevant points.
To modify the expression for your own use, simply change the layer reference and the map units from 20 to 1. Make sure both layers are in a projected CRS (miles-based in your case)!