The simple answer to that is that there are many devices and associated bits of software / web-based services which will allow you to do all those things using one device. The tricky questions are around what your priorities are, the relative importance of each activity, and the usability of the device whilst doing that particular activity. There's a reason (beyond marketing!) why there are numerous types of devices, and that's because there is no perfect solution, using one device, for all scenarios, and that essentially comes down to the primary considerations (arguably) of a) screen size, b) 'mounting', c) battery life.
(For background, I've used portable gps receivers since 1995 and all sorts of different devices over that period, variously up mountains, on foot and wheels and for short (hours) and long (days) periods.)
It's pretty much a truism that for any given activity (cycling, hill walking, driving, whatever) the piece of kit designed specifically for that purpose will be 'best', meaning 'best combination of compromises for that particular activity'. A bike computer is designed to be mounted conveniently and visibly on a bike and will have bike-specific functions. Conversely, again taking a bike computer as an example, it's not the best choice for hill-walking due to a combination of ergonomics and functions, but it'll work OK. If you have a hand-held (walking / mountaineering biased) gps receiver, it'll do the things you want to do on a bike, but the screen will typically be smaller and mounting might be fiddly (to say the least). If you're running, pretty much anything of handheld gps or bike computer size is going to be a real pain due to the weight and bulk, even if you're merely recording and not looking at it whilst running (but nonetheless it would be usable and could do all the things you specify).
If you prioritise displaying actual OS maps on a screen (I'd not rely solely on a device for walking/mountaineering and would also carry map + compass), then you're stuck with needing a device of handheld gps or bike computer size, since you need the 'big' screen. If you can tolerate having rudimentary maps which will show you a pre-planned route, including direction indication and progress along it, and then back that up with paper maps in case you need them, a gps watch covers everything. i.e. the only functional thing a gps watch won't do is display a proper map on a screen. Ergonomically, the watch works pretty much best for all your scenarios bar cycling, where you can't readily stare at your HR or cadence or whatever (safely!), but can refer to it (of course, you can wrap it round the bars, so that issue can be overcome too if it's important.
So what I'm getting at here is to first make a list of priorities and weight them, and those priorities should be all about ergonomics since many devices can do everything you have listed, but it's the ease of use which varies. For what it's worth, I have quite a few devices accumulated over years and in most scenarios I solely use a 'high end' gps watch these days. The pure accessibility wins easily over handheld devices when on foot, and to me the compromise on the bike is insignificant over a dedicated bike device (and having just one device for everything is, again, to me, a big benefit. (If I was doing serious training on the bike and wanted to stare at power and cadence then I'd use a bike-specific unit, but I'm not, so I don't!). I've used a gps watch, in the last year and without issues, for all my cycling, alpine mountaineering, hill walking and a pretty long fell run, in all cases not referring to the backup paper maps I [usually] had with me for emergencies, so that's my 'best fit'; it might not be yours though.
Once you have really thought about what type of device (watch v handheld gps v bike computer) then you can just narrow down your search to units of that type, which makes it a lot simpler and quicker, taking into account the specific functions you [expect to] need/want.
