Unguided dumb kinetic projectiles have been used in wars gone past, such as the "Lazy Dog" used in Vietnam. You just shovel loads of them into the sky over the target, and let their kinetic energy take them through any cover (such as light roofing or tree canopy) and through any soft target underneath. To quote from the wiki page on kinetic bombardment,
These were dumped from aircraft onto enemy troops and had the same effect as a machine gun fired vertically. Observers visiting a battlefield after an attack said it looked like the ground had been 'tenderized' using a gigantic fork. Bodies had been penetrated longitudinally from shoulder to lower abdomen
(original sources are books, and the google book links don't work for me)
One of the dart shapes could apparently penetrate over 60cm of sand, which is quite effective for such a small projectile. Similar weapons (called, more generally, flechettes) were used in WW1 and 2, though I don't have any reports on their use or effectiveness.
If your swords fall lengthwise, either the pointy end first or a suitably sharp pommel (not that such things are very common), then the effects will be fairly similar, though swords are generally a bit bigger and heavier than flechettes or lazy dog projectiles.
Rain generally falls from 2000m upwards. Lets say a sword ways 1.5kg. If it falls from that height it'll be going at about 200m/s, giving it a kinetic energy of nearly 30kJ (neglecting air resistance, but swords and skinny and dense so it shouldn't make too much difference. Probably). This is nearly twice the muzzle energy of a 50 cal machinegun round, but with an awful lot more momentum due to its substantially greater mass. I'm not totally certain what kind of penetration this will have in most materials, but investigation of the effects of 50cal MG fire might be informative. Velocity of the falling swords scales according to the square-root of the height they fall from, so their kinetic energy scales linearly with height. Some storm clouds can be very tall... I'm not sure what the terminal velocity of a slender bit of falling steel is, but it is likely to be fast so this 50cal comparison is probably at the lower end of destructive potential.
That just leaves the how many swords fall. A seriously heavy rain shower might drop 50mm of rain in an hour. Over a 1 metre square patch of land, that'll be 50l of water. Given the same mass of stuff falling, that's 100 of the aforementioned swords for every 3 square metres of land. If your plot of land is slightly larger than 5x5 metres, you'll get enough swords every year to make your very own iron throne. Melting the metal bounty down might be tricky though, because all the trees will be gone, so no charcoal or wood for cooking or forging. Maybe you'll have access to coal, and can use the swords as mining tools.
As well as the trees, this bombardment will also destroy most light constructions (such as houses), erode rock, churn up land and ruin crops, sink boats, slaughter livestock, etc etc. It won't take many years of this happening to destroy most large woodlands and macrofauna. If it has been going on for a while, trees and livestock and horses and so on simply won't be things that exist anymore, except in carefully maintained armoured environments. Steep valleys will fill with rusting iron. Rivers will turn read and brown, and might poison fish (those that haven't already been shredded, so probably only the little ones) or at least cause dissolved oxygen reduction. You might get sword avalanches on very steep hillsides.
The UK's current area of land devoted to agriculture is about 93 billion square metres, so in order to keep the land useable (lets not think about the fact that anything growing or living on that land might have been ruined) you need to shift some 4.65 gigatonnes of steel. With about 55 million adults in the country, they'll each need to shift about 85 tonnes of steel. I don't know where they'd put it all, though. You could send it out to sea, but all the boats will probably have sunk...
Sword facts and figures were added after I did my calculations. They end up being even more bad news than my example. 3600 swords per square metre per hour instead of 34, 9 tonnes of steel instead of 50kg. I won't bother working out the coverage of land, because it would be pointless. Everything gets buried and crushed to death, even if it doesn't get stabbed. Underground facilities have to tunnel their way out through a layer of steel to escape.
Oh, and projectile kinetic energy now nearly 50kJ instead of nearly 30. Swords fall, everyone dies.