Get way from location

Several options, all of which require that you set up your own server:

• load OSM data into PostGIS database with osm2pgsql (ideally configured to load only roads), then make ST_DWITHIN query to find all roads in the vicinity of your point and order them by ST_DISTANCE. This will give correct results even if your query point is a long distance between road nodes.
• set up an instance of the Nominatim geocoder (also requires server with PostGIS etc) and have it reverse-geocode the position you are looking for; since this will occasionally not return the road itself but also POIs or houses, toy with the "zoom" parameter or modify the software to only return roads

Options that do not require setting up your own server, but that will get you blocked on the respective servers very quickly if your app sends these queries every couple of seconds:

• use existing Nominatim servers and reverse geocode your location (of course the "modify the software" option is not available then)
• use Overpass to request all "highway" type ways in a rectangle around your point, with their nodes; then, as you describe, compute the line segments between consecutive points in these ways, and find out which of these line segments is nearest to your position.

Neither of these options will correctly find out whether you are on a bridge or under the same bridge - this is something that would require additional analysis on your part (I was there 10 seconds ago, now I am here, this means I cannot be traveling on that road, it must be this road...)

Do you mean such query? The location I assumed.

/al