How widely used is the word "tush"

It's a little difficult to say how widely it's understood, but I can say this: Whenever I've chosen to use that word, no matter who the listener is, they understand me. No one has ever said to me, "What's a tushie?" On the other hand, maybe context has always been effective in clearly implying the meaning.

As for what words others use instead, I imagine you can find a fairly comprehensive list in a number of locations, but I think what you're really asking is what euphemisms are used. Some of these are:

rear, rear end, seat, bottom, bum, seat cheeks, tail, behind, derriere, caboose, booty, trunk

And the list goes on.

In New Zealand English it's not used, but given reasonable context it might be understood (ie the meaning guessed).

The nearest equivalent would be "bottie" - we have a lot of diminutives formed with "-ie" like this.

Examples: "Nourishing and soothing, "Sweet As" baby's bottom balm helps keep your baby's bottie sweet as." - http://www.historicplaces.org.nz/shoponline_2013/healthandbeauty.aspx?sc_lang=en "Great for quick nappy changes, with a generous bottie area allowing for bulky nappies, these newborn pants look great..." - http://www.thebabysroom.co.nz/category/pants