Oh, my, you do have a problem! But not all is lost. You can fix this. I once had wood develop algae almost as bad and was able to clear it up surprisingly quickly.
While H2O2 (hydrogen peroxide) will kill BBA, I would not rely on it as your sole source of attack when the BBA has gotten this bad. The best way to approach this would be to first clean it up manually. It's a pain, but it's the best way to go.
Anything that has BBA on it that you can take out of the tank, remove from the tank and scrub off as much BBA as you can physically remove. Scrub every inch of it. Leave as little BBA as you possibly can.
Then soak anything non-porous (such as rocks, intake pipes, spray bars) in bleach water for a couple of hours. A 1:20 bleach:water ratio should work fine. I generally just fill up my sink with water and pour bleach in without measuring.
Anything porous (such as wood) spray with H2O2 and then soak in H2O2 water. Leave it there for 30 minutes.
After the items have soaked, scrub them again. Often, dead algae will scrub off easier than living algae so you should be able to remove the last remnants of algae that wouldn't come off before. Use any stiff bristle scrub brush you can find that's not full of soap. I've also used magic erasers
to clean intakes and spray bars to perfection. Once everything is clean, soak anything that was bleached in water with a high concentration of dechlorinator to remove any trace of chlorine that may remain.
For stuff that you cannot remove from the tank you will have to scrub off the BBA while it's still in the tank. I've done this with wood in tanks before. Yes, it makes a huge mess, but you'll clean that up. You can remove your fish if you'd like, but I never have, and I've never had any problems with my fish, inverts, or snails having done it (however, I wouldn't suggest this for sensitive inverts, such as CRS).
First, turn off your filters unless you're willing to clean your filters when you're done. Otherwise, the filters will get clogged very quickly. Then take anything you can find to scrub off the BBA from the wood. I've used scouring pads, scrub brushes, and toothbrushes. It's a time consuming process, but you have to do this. Scrub every inch of the wood. Get all the BBA off of it. ALL of it. It usually takes me a few hours to do this, but it's well worth the effort. The results are amazing.
Once you've cleaned off all the BBA from the wood, do a water change, doing your best to get all the BBA out of the tank. If you have to do more than one water change, then so be it. You can start out by using a small net to grab as much of the large pieces of BBA floating around which will reduce the need for multiple water changes. I've also used a large plastic spoon
to "swoosh" the water around plants and the substrate to kick up debris into the water column so it can be netted. When you think you have netted all you can get, then go on and do your water change(s).
After you've finished your water change(s), turn your filter(s) back on. They should finish cleaning the water. The next day, change out your filter floss as it should be clogged.
NOW it is time to start treating with H2O2. This will kill out the small amount of BBA you could not get off the wood and decor. Use 1ml of H2o2 per gallon for your tank (so 40ml for a 40g tank). Turn off your filters. Use a syringe to squirt the H2O2 directly onto the BBA. Once you've used up the full 40ml, turn off your lights. Wait 15 minutes. Then turn on your filters. Wait another 15 minutes. Now turn on your lights. You can do a water change if you want, but it's optional.
I have used higher concentrations of H2O2 (as much as 4ml per gallon); however, the higher you go, the riskier it becomes. If you have done a good job of scrubbing off the BBA, you should not need to take the risks of using higher H2O2 doses. You can repeat the H2O2 treatments as often as twice a day, but space them far enough apart to be certain that the first treatment has completely broken down (H2O2 breaks down into water (H2O) and oxygen (O)). I generally wait 9-12 hours between treatments.
The more algae you kill with H2O2, the more you will need to keep up with water changes to remove the dead algae from the tank; otherwise, the dead algae will decompose and turn into ammonia which can cause your tank to mini-cycle (another reason why physical removal is preferred over using H2O2 with a bad outbreak of BBA like this). So it's always a good idea to test your water for ammonia, nitrites, and nitrates while you're doing all this to be sure your cycle remains in good shape. If you do get a mini-cycle, perform water changes as needed to clean out the dead algae and to keep the water parameters in check. Dose with Prime every 24 hours to protect the fish from any harmful effects of the mini-cycle.
It is always best to catch an algae problem before it gets this bad, but it's not impossible to fix it once it's taken over like this. While my suggestion may include a lot of hard work and may make the tank look disastrous while you're working on it, the results are excellent. The two times I've done this, it has completely resolved the algae problems after just one round of cleaning. So the work is well worth the effort and much preferred over spending months and months of fighting the algae. I've even won the battle over a combination of BBA and Clado using this technique.