First don't let anyone stand near the work area as delapidated gutter can fall on mass, I've experienced it in 20yrs reparing /replacing guttering.
Well start with OGEE which is flat backed and fancy gutter front. Screws hold it on facia at back.
1. Whack a 150mm nail into facia under collar of joint your working on for support when screws and collar bolt free.
2. Cut off any bolt surplus stick through the nut so when undoing nut it hasn't got far to go.
3. On the next section that's sitting in the collar you've supported release the first back screw.
4. Gently tap screwdriver between the joint from underneath. wedge up the spigot section gently and not too high hold in place with another nail. clean out the collar and tap gently any putty on the spigot. Re-join using putty for metal, press two together and fit new bolt. tighten but don't over do it job done rescrew backs job done.
Half round forget the support nails if the brackets are in good nick. same as above for prep of joint.
Post how you get on.
rob
May 2007