First you need to put a clean sheet on the mattress the hard way. That one is going to stay a while, so it needs to be one you don’t mind seeing the side of.
Get a mattress topper
Then you need to get a mattress topper of sorts: an egg crate, a memory foam pad or something of the sort.
Put the mattress pad on top of this smaller topper, and voila. Easy changing of sheets. You can either lay up there and bend up the corners or you can pull the whole thing down if you prefer. The top sheet and bed spread are still difficult to make look nice, but that doesn’t bother me.
There you go. Cheers.