In the UK this is controlled/coordinated by the BMC.
They publish an extensive guide on how and when to do this Bolt Guidance Document.
There is also a working group, coordinator and they even pay for new bolts. There are lot's of issues around this, the document goes into a lot of detail these. The most important one (in the context of your question) is.....
When to replace bolts
(sic: Bolts do fail but not very often).....After all when was the
last time you heard of a bolt failing? Well it does very occasionally
happen and recorded cases suggest in some situations that are hard to
predict. So what factors should we take into account?
Rock quality
Like any other form of protection, a bolt is only as good as the rock
it is placed in.....Always remember that the erosion that creates our
crags has not halted. The weather and particularly cliff vegetation
has great capacity to loosen and lever off even very large lumps of
rock. Once sound placements have been known to change as the rock
around them unexpectedly crumbles.
Depth & Position of Bolt
Bolts placed close to edges will have a reduced strength; 200mm is
often quoted as a minimum clearance distance. Unfortunately bulging
limestone doesn’t usually accommodate bolts being placed with that
allowable distance. Likewise, bolts placed close to cracks, pockets
or other discontinuities may well be reduced in strength.
Evidence of Corrosion (Rust)
....Just because a bolt is rusty it doesn’t mean it is dramatically
weakened. However, if it is rusty it has lost some metal and
consequently some of its holding capacity. It is not unknown for the
nut to be of a different metal and to have corroded so that it no
longer holds the hanger in place. Bolts where the hanger is fresh and
shiny but the bolt itself is rusty may suffer from a particular form
of corrosion and have been known to break with the lightest of load.
It makes sense to inspect bolts that are placed in drainage lines.
Unfortunately the wettest place part of a bolt is in the hole and
behind the hanger making inspection difficult.
Accidental Unclipping
Hangers or eyes ideally should be of a size that they don’t allow the
karabiner to snag in such a way that the gate can be forced open and
unclip. Correct orientation of the karabiner gate away from the
direction of travel also helps avoid this.
Spinning Hangers
Hangers can spin because the bolt has moved in its seating or because
the securing nut has become loose. If there is evidence of the former
then the bolt is likely to be unsafe. If the latter is the cause then
the nut should be immediately retightened. The leverage and wear
caused by a weighted hanger rotating on the bolt body can give rise
to weakening of the unit due to the formation of microscopic
fractures.
Old Bolts
Despite the number of renewal programmes there are still many old
bolts out there, especially ones used to protect blank sections on
80’s trad routes. Common sense dictates that bolts have a limited
useful life and it is always worth consulting the guidebook to see
when the route was first done or, if the gear has been replaced, when
that might have been done. Whilst stainless steel products are likely
to be good for a lot longer, experience has shown that many bolts
placed in the past are dangerously weak after 10 years. Some early
staples had no notches on their legs making them prone to pulling out
relatively easily if an outwards force is applied.
Wobbly or Damaged Bolts - Mechanical or Glue-Ins
It makes sense to doubt the holding power of any bolt that wobbles in
its hole! With glue-in bolts a problem can arise if the drill dust is
not thoroughly cleaned out of the hole or if it was placed in a damp
hole. Some wobbly glue-ins have been found to take quite high loads
but it’s probably not sensible to test these with your own weight!
Unset Glue
Glues have to be mixed properly. Failure to do as instructed can
result in the mixture not curing, or hardening as was designed and
not holding the bolt in place. Several serious accidents have
happened when climbers have come across new bolts and attempted the
route without checking the glue has set.
Deteriorating Glue
Like the bolts themselves the glue can and will deteriorate with time
though for appropriate glues this should be a very slow process.
However, there is anecdotal evidence that some inappropriate glues
have been used, that are not suitable for the alkaline solutions
likely to be encountered in limestone rocks. Unfortunately this could
mean that once sound bolts become less so, perhaps as little as two
years after placement.
Worn Belay Bolts
If top-roping a route the climber can always arrange slings so that
both bolts are weighted. This is good practice and climbers should be
discouraged from top roping with the rope directly through the belay
bolts, as significant wear on the eye can quickly result.
Home Made Hangers
Various types of bolt hanger were homemade during the 1950s and 60s.
Usually these were simple angle iron (or aluminium alloy) with holes
drilled in each side. A few may still be encountered on old aid
routes today but generally they are now of little relevance and most
have been superseded by modern bolts or other forms of protection
Stress Corrosion cracking
Although there is no evidence of this type of attack occurring in the
UK, a number of bolts have failed by stress corrosion cracking (SCC)
in Thailand and Cayman Brac so it’s worth being aware of it if you’re
going sport climbing in the tropics. SCC can occur in stainless
steels in aqueous chloride solutions (e.g., seawater) so the bolts on
sea cliffs appear to be most susceptible. Luckily for us, there is a
temperature (~50°C) below which it doesn’t occur (except in very
acidic conditions) so it shouldn’t be a problem in the UK until
global warming really kicks in. The corrosion is very localised and
takes the form of cracks that can penetrate through the metal,
reducing its strength to almost zero. These cracks can be very fine
and difficult to detect on the surface and impossible to see how far
they penetrate
There is also a part II. This goes into details on how to go about actually replacing the bolts.
When it comes to removing the bolts, it depends....You can sometimes unscrew them, sometimes the need chopping. TBH, it doesn't matter as you should drill a new hole, etc. for any new bolts anyway. Just make it obvious that the old bolts should not be used/remove them as best you can.