Change elevation units from meters to feet?

Is that some sort of app? There's nothing by that name in the Android app store. The Apple app store shows something nearly matching that name as an in-app purchase for a third-party app. If your question is about a third-party app you will need to contact that app's developer.

If you're asking about maps available on the https://www.openstreetmap.org/ website, then they're the same set of maps for everyone in the world. They're optimised for the 99.5% of countries that use SI units rather than the 0.5% that still use Imperial.

I have now downloaded multiple versions of this map.

In looking around I can see there are multiple organizations that provide or use OpenStreetMap

While most use meters, I was able to locate a provider or two who uses OpenStreetMap as a base with one key difference, all of the contours and elevations render in feet. That, for me, is important. The ability to choose which system to use and have it render correctly.

In comparing the two maps (OpenStreetMap - World - which renders in meters in spite of the setting stating it is rendering in feet) and another OpenStreet World version which accurately renders in feet, I have found the topographical features are far more accurate when rendered in feet. I took a sample of some of the local trails and mountains and compared them. In terms of accuracy the version using feet was far more accurate. The difference is not funny. I am looking at them now. The version rendering in feet is accurate

In looking around I can see there are multiple organizations that provide or use OpenStreetMap

Yes, that's correct. Where did you find the maps or apps that met your needs? Do you have a URL for them, or a name in an app store? I'm not even sure whether you're looking at a website or an app at this stage.

Can you give the actual names of the apps that you have looked at or tried.

Out of curiosity I played around with the iPhone version of OsmAnd and found that even if set to feet/miles it still shows mountain peak elevations from OSM in meters. This behavior does not seem to be correct. But it is an issue with OsmAnd not OSM (different projects altogether though OsmAnd uses OSM data).

Regarding "topological features" by which I think you mean contour lines, that data is not in OSM and a map renderer or app developer has a number of different choices on where the get the data. The easiest to find is the older Shuttle Radar Topography Mission (SRTM) data which is fairly course and will result in contour lines that are not as good as they might be.

Guru Maps (formerly known as Galileo Offline Maps) for Offline Maps & Navigation. Available on the App Store and Google Play. It uses vector Maps based on OpenStreetMap data. I know because I added data to OpenStreetMaps and it is now showing correctly in that application.

The version on the iPhone for TopoMaps - I love the OpenStreetMap World version map with the noted exception of the fact that it renders in meters and does not accurately place the summit of some of the peaks. The application being in meters gives the false impression that a couple of these peaks are easy walk ups. A gentle stroll up a hillside. The reality is quite the opposite. They are not easy walkups- they have cliffs in several areas and there are some very steep sections. The rendering in feet with the contour lines and shading puts them into their proper perspective. Not quite as accurate as USGS topos but far more accurate than the rendering in meters - I am downloading a few additional maps to compare as well.

When I go into settings

Where? Is this on your PC, or phone, or somewhere else?

Apparently, it does rather more than I realised. https://en.wikipedia.org/wiki/United_States_customary_units#/media/File:Metric_and_imperial_systems_(2019).svg suggests that the US is something of an outlier though.

OpenstreetMap Data is just data. If you want to create a map using other units (rods, hogseads, whatever), you are entirely at liberty to do so.

Elevations should be in feet in the United States. That is the system US Citizens have grown up with and are familiar with. It is also the unit of measurement that is used on a day to day basis. Meters are not helpful. If you want to make OpenStreetMaps better then update it so that it uses the local system of measurement.

A native United States citizen here, raised on using the old USGS quadrangle maps that used the NAD27 datum and displayed elevations in feet with 40 foot contour lines. In my not so humble opinion, there is no reason that a website that serves the entire world should tailor its display using the units used in only one country in the entire world. Especially since the main map at https://www.openstreetmap.org/ is simply a demonstration of a possible renderer. There are many different renderers tailored to different applications and the "default" one put up by OSM does not render everything.

If you want a renderer tailored to US expectations, there are some. See, for example, https://wiki.openstreetmap.org/wiki/OpenStreetMap_Americana and https://wiki.openstreetmap.org/wiki/TopOSM

There should be an option to display in either meters or feet - according to ones preferences or the option to edit the elevation of various peaks to show both feet and meters. When I use Open Street map, I ask it to display my settings in feet. The map however displays in meters inspite of that setting. The expectation is that the user should be able to see the map render in either feet or meters. In the US, feet is the system that is used on a day to day basis. This should be not that difficult of an improvement. It would be extremely helpful.