The ODbL does indeed require that, if you add to OpenStreetMap data, you have to make that data freely available under the same terms - that's the quid pro quo for getting access to such a huge amount of geodata. It's what's known as a "share-alike" licence. I'm afraid that 'free-to-view' generally doesn't have any special bearing under open source licences - you'll often hear people make this point with the phrase "free as in speech, not free as in beer".
If you want to build on a UK map database but not share the results, you'd be better off looking at the Ordnance Survey OpenData datasets. These have no "share-alike" requirement, so you can do whatever you want as long as you retain a credit to the Ordnance Survey. Their main datasets are Meridian2 (low-resolution street data, probably the best for your purposes); VectorMap District (may be relevant relevant, but unroutable, partly only partially attributed, and possibly too high-resolution); and StreetView (a raster product so less relevant).