I need to replace the circular blade on my Ryobi EMS-1425L Mitre Saw. How do I lock the blade to undo the securing nut (can't find the instruction manual)?

it may be worth investigating if later driven and pinion gears will fit the shafts of your unit
It is a way that ryobi has of making older unit obselete
if you research the gears by shaft size , number of teeth diameters thickness etc you may get something but look for matching gears and not just the stripped gear
you may get gears from another manufacturer/model to suit

Is this the saw you are talking about? A compound mitre saw?
TFO 250 Ryobi
I'm assuming you cannot lift the blade up by the handle without the whole saw coming with it. That usually means that the saw is locking into place for transportation purposes. Look at the pictures below and see if the lock pin in engaged.

Have you checked on Ryobi? You might want to check Kreg fence. What type of blade are you using right now? I have a saw and I got online supplier for blades , I purchase on sawblade.com , happy with them. Good luck.

http://www.ryobispares.co.uk/assets/applets/CW1440.PDF

Most times when I need to change a blade on my 7 1/4" saw I simply grab the blade on the front side, below the saw's support table, with a pair of channel lock pliers and then use the box end wrench on the blade nut, pushing down from the back side of the saw blade. That should loosen the nut so you can take it out with your fingers. The reverse of this procedure should tighten the nut on a new blade.
I'm not real familiar with the Fury saw, but if it requires saw blades with the diamond shaped cut-out in the center, most of the older blades had a knockout that will fit the diamond shape it would need.

With any circular saw, table saw, metal cut off saw, mitre saw etc the nut always tightens against the direction of rotation and loosens the opposite way

I can't find that model number listed but generally a circular saw is standard thread, the mitre saw us left hand and a worm drive is also left hand thread.

It should still easily fit a 5/8" blade. The arbor shape shouldnt have any effect on the type of arbor hole needed on the saw blade (only round arbor holes are made on saw blades. You will probably find the arbor securing nut has a shape that fits into the flat gaps between the saw blade and the arbor when fixed. So grab a 5/8" round arbor saw blade and give it a go. It should work fine.
It is the arbor nut that secures the blade onto the arbor via friction which holds it on.

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There may be a push button type blade lock, and on many of these saws the retaining bolt for the blade is backwards threads, e.i., looking at the top of blade retaining bolt, turn it clockwise to loosen. This would normally tighten, but in the case of many of these saws, it loosens it.

unplug saw
new Ryobi have a spindle lock button, it is unobtrusive and you may miss it
If yours does not, secondhand etc, a screwdriver in the blade teeth between the upper and lower guards provides a locking bar enough to undo the arbor bolt