Your iPhone (5c) is not compatible. You need an iPhone 5s at least for any Apple Watch and an iPhone 6 in some cases.
No. The iPhone 5c is compatible with Apple Watches (first gen, series 1 and 2) running watchOS 1 through 3. It's only Apple Watches running WatchOS 4 that require iOS 11 and as such an iPhone 5s or later for all watches except the Apple Watch Series 3 GPS+Cellular which requires an iPhone 6 or later.
If the Apple Watch is still running watchOS 1 as explained by the OP, which only the first generation Apple Watch ever ran, then it should still work with an iPhone 5c, as it did before watchOS 4 was released.
You only need an iPhone 5s or later if the watch is already running watchOS 4 or you want to update to watchOS 4.
The real question is whether watchOS 3 is still available to update to. Since all watches support watchOS 4 at this point Apple may have removed watchOS 3 at this point and that may be the reason no watchOS update is available for the watch when paired with the iPhone 5c.