OK.....
First, please can you give us the overall length in mm and the "sounding length", from the centre of the embouchure hole to level with the far, open end. Those will let us confirm the pitch diagnosis fairly well. I'm betting on High Pitch Bb (i.e. an instrument nominally in Bb but built for the old English higher pitch standard of c A=456, not 440 - which would make it a tad flat of being "in B" at 440, FWTW.). You can also confirm this by checking (if it will sound) the 6-finger note (all holes closed) against a piano - if it is near as damn it a B, your toot is HP; if you're lucky, it'll sound a Bb and you'll have a (usable at) concert pitch example.
(Youse guys who've commented above, put your specs on - youm slipping!
)
Cleaning.
(I take it you're in GB, anoia? Where by?)
OK, it doesn't look too bad in the pics (compared to some I've seen!). I don't think it'll need washing. I have done that recently with a very fine old flute which was in chimney like state from old tobacco and beer residues - completely stripped it, then rubbed the wood all over with washing-up liquid, scrubbed the exterior with a soft nail brush and the interior with a bottle brush, then rinsed and scrubbed in warm water - repeated a couple of times but not allowed to soak, let dry, then cleaned with meths. It came to no harm and is playing just fine, back with its owner. This case does not appear to need such drastic measures.
Go buy some meths - a 1/2 litre will cost under £2 in any hardware shop. You won't use all of it, but it keeps indefinitely and is a useful addition to the under sink cupboard or DIY shed, like white spirit (don't use the latter - it wouldn't harm, but it stinks!).
The "short F" key is probably only sticking because of electrolytic corrosion products and stiff, old grease/oil in its axle, but it is possible the spring is tired or cracked. Worry about the latter later.
Take the keys off - make sure you keep their own axle-pins with them individually, but you shouldn't have any difficulty identifying which one belongs to which hole/pillar set because usually they won't fit in the wrong place. Push the pins out a few mm from the side which looks to have the tidiest end (usually one end has been filed smooth/round after being cut off) - use the tip of a small cross-head screw-driver or some other suitable probe. If they are very resistant, try from the other side, though often one end is left slightly burry or is pinched/spread and won't push through. If they still won't shift, you may need an extra pair of hands to hold the flute and something to gently tap the probe to start the pin; or set up a way of squeezing it in a vice if you have one. If such measures are required (probably not), obviously be very careful not to put too much force on the pillars, which could split the wood or tear out. Once the pins are sticking out a few mm, you should be able to twist/pull them out with the nose of a pair of pliers - preferably pointy ones to avoid scratching the wood.
Once the keys are off, you can clean them thoroughly with Brasso &/or metal wadding. Use those also to clean all the other metal parts, including the pillars, before you clean the wood. If the springs will rotate on their rivets, you can clean them and under them too. It will be best to remove the (apparently dead) old pads before cleaning, they - don't look worth saving. Do that by heating the back of the key-cup with a fag-lighter to melt the old shellac or sealing wax which is most likely holding them in (if they're glued, it should help loosen that too) - hold the keys with your pliers if you don't want burnt fingers! The pads should sort-of bubble up and then you can flick them off with a knife or screw-driver. Scrape the residue of adhesive out of the cup and then do your cleaning. Of course, you'll need to get replacement pads - of which more later. Remember to clean the (probably steel, but the wadding or Brasso will do fine) pins and to use a pipe-cleaner to clean the inside of the axle-tubes and pillar-heads. If any of the ferrule rings come off/are loose, don't worry - just tell us.
You have no tuning slide, and I suspect the head will be unlined (no metal tube inside), but you have a stuck joint? You have to get this apart. I thought there was some advice on this on
Terry McGee's website (an invaluable resource), but can't (quickly) find it - and I know there are old discussions of it here on C&F (use the search tool). Inside the joint will be either thread or (less likely) cork lapping which will be perished and probably not be secured to the tenon any more and is rotating as you turn the head relative to the body, but bunching up when you try to pull them apart. You have somehow to work them apart until you can (if necessary) start to pick away at the dead lapping with tweezers or a knife point. I'd try using the back of the blade of a knife in the crack you have between head and body and just very gently twist-levering them apart, working around the circumference and not rushing for too swift progress. Try to lever against your fingers, not the shoulder-edge of the body, as that will easily chip. One often finds that, once a few mm have been gained, you can twist the sections apart in the normal way. Don't worry about the old lapping - it will have to be replaced anyway. Do be very careful not to over-force anything, though - you could easily crack the head socket (if it hasn't already got a crack). Patience and persistence!
Once you have the head off, you will need to remove the crown cap and push out the stopper cork (they may, but probably won't on this level of flute, be a screw-adjustable, connected unit) - probably easiest to get a length of dowel (wooden spoon handle will do nicely) which will fit loosely in the bore and push both stopper and crown out from the socket end. There's a good chance the cork may be perished and crumbly - don't worry about that either. Once they're out you can clean the metal parts with Brasso/wadding and then you're ready to use rags and meths to clean the wood. A thin dowel (6mm/1/4") is useful and old cotton rags from dead sheets/pillow-cases are ideal. I usually wrap a rag around a dowel and push a loose amount into the tube from one end, then pour in meths from the other (over a bowl or some such), or you can dip the rag in the meths first. Give the bore a good swabbing (you may need to scrape away cork residue in the head if the stopper cork left bits stuck to the tube walls) and rub down the outside with the meths-damp cloth, attending to any detail with finger-nails through the cloth. Use cloth wrapped around the dowel to clean the embouchure and ditto with something smaller - e.g. a knitting needle - for the tone-holes. Repeat as necessary until you are satisfied.
That's enough for now. Once you get that far, let us know and we can address the re-assembly process and such matters as replacement cork, pads, lapping, oiling and any repair issues that arise, like cracks, broken or breaking springs, loose rings, etc.
To encourage you, once it is clean and has lapping and stopper cork, you can at least stop up the keyed holes with blutack and play it!
Lastly, if you're anywhere in reach of me (N.E. Wales), I'd be happy to help more directly.