The wood that I used was an old curtain pole that had been in my neighbour's garden for about two years; it was rotten at the ends but that was no problem as I only needed about 8" of it. ( I chose it because it was just a little bit wider diameter than my handlebars; I also looked through my firewood but nothing was uniform enough.)
I used UHU contact glue. (Update: I used Two Part epoxy resin glue because I wanted to be sure of the grip).
The tools that you need are a drill and a 22mm spade bit.
(A drill press would be good and a lathe would be untold but I have neither of these.)