For mattress protectors, use wool. Wool works great -- you can cut it to fit; no need to hem even -- and it works very well for repelling moisture and bacteria growth. And not all mattress pads/covers are polyester-based. I found an all-cotton one at TJ Maxx for cheap not long ago, actually.
The thing about wrapping mattresses are some potential issues to consider:
1. You are sleeping on plastic. True, it's a foodgrade plastic that doesn't offgas, and it locks in the chemical gasses of the mattress, plus it's waterproof, so that's good, but it's still plastic, which means it's loud. You have to have some kind of protection between it and your sheets unless you want it to feel slippery and be loud underneath you. Thick wool would be great for this, I'd think.
2. Because it's totally encasing the whole mattress, it's potentially trapping in moisture. Which can result in mildew issues. This, I would think, would be tied into whether your home is older and/or damp, etc. to begin with? I don't know. Something to consider? It was a concern for us but we live in an older house.
3. If it's a partial wrap (top only) then you're still dealing with problem #1 and there is still off-gassing from the mattress itself. I'm pretty sure the Bebe ones are full wraps though, yes?
Whichever wrap you go with, make sure it is 100% foodgrade polyethylene. You can get one for a larger bed (if you do end up cosleeping and want to wrap your older mattress) here:http://www.offgassing-mattress-wraps...ress-wraps.htm
Get a wool cover for it. This can be anything from a wool blanket (like from the Army/Navy store) to a full on tufted type mattress pad:http://www.allergybuyersclubshopping...ads.html#sizeshttp://www.snugglewool.com/shop/catalog.php?category=17
(this one has bassinet sizes)
The thicker the pad, the more protection in terms of sound and feel from the plastic.
If money's an issue, look at the total of a new mattress (non-org), wrap, and wool pad versus organic mattress.
We got this crib mattress from Dax:http://www.daxstores.com/v-snq-crib.html
(they have free shipping)
It was affordable. The twin mattress we're getting for G's new bed is much more expensive. But still barely over what we'd spend for all the others. So it's a tossup.
Also, be aware that you may have to get custom crib sheets made to fit if the mattress/wrap/pad ends up thicker. Our organic crib mattress was really hard to fit, but we did find some that would.
In terms of your bed being older, and your pillows, etc, if you're really worried, you can wrap the mattress and replace the pillows. Apparently pillows should be replaced every year or two anyway, esp. if you don't wash them regularly. Using more than one pillowcase at a time (like the zippered kind underneath) can help I think, as can washing them every 6 months. Be sure to let soak first and run extra rinses. Make sure they're totally dry before sleeping on them.
There's one school of thought that older pillows/bedding/mattresses don't offgas as much and aren't as dangerous. Other people believe they are more harmful because of the accumulation of bacteria and microscopic bugs, etc.
I'm of the former. I think that if you keep your pillows and bedding clean, and have some kind of mattress pad on the bed, that helps. And that overtime while the chemicals don't all dissipate, I think that they are less harmful. We have an organic crib mattress, but our mattress is not organic and we coslept a long time. The pillows on Winnie's bed are organic or all natural. My pillow is memory foam (which doesn't smell anymore but is still probably bad). Her bedding is cotton but she also has fleece blankets. I don't have issues with plain polyester. I have issues with certain types of poly-fibers that offgas. Not all do. It's a personal thing. If you don't like it, don't use it.
The problem with used baby/children's bedding is that you're looking at different fluids potentially on/in them and resulting moisture and/or bacteria there, in lieu of or in addition to any chemicals that may still be offgassing.
I would definitely replace the bassinet mattress, if it's pre-used. And there are some things I would never want near the baby -- including vinyl/PVC, which is the typical cover for baby mattresses. You can make a mattress for the bassinet if need be, out of wool and cotton quilting batting. It wouldn't need to be more than 2 or 3 inches thick. Just make sure it's firm and the bedding fits tightly. Be sure to keep the room no warmer than 68-degrees or so and don't overheat baby with too much clothing. Those are contributors to SIDS too. Always put baby on her back to sleep. I found I couldn't sleep without baby right next to me. And she couldn't sleep unless she was right next to me. There are lots of cosleeping links out there that show it helps prevent SIDS. I'll send you some if you like. We didn't plan to cosleep but did and I'm very glad we did. We plan to with new baby.
I can understand the concern about avoiding any non-natural fibers, but if your sheets are a blend, and are so old you don't recall if they're a blend, my vote is they're fine.
That's just me. But as with anything, if you're not comfortable with any polyester, then there are plenty of ways to avoid it. Wool and cotton are your best bets, for sure.