Buds in this context means 'nodes' the positions where leaves are/were.
If you look just there, at each node you will see a small (often red)
'bud' - sometimes quite flat, sometimes prominent. This is where you'll
get new growth if you prune just above it. With roses you normally get
new growth from 2 or 3 nodes below the pruning cut. In old wood you
can't always see the nodes but if you are in radical pruning mode you
can often get away with making a random cut which may or may not force a
dormant bud to shoot. Roses are hard to kill but please remember the
type of pruning you're proposing is a 'kill or cure' treatment - mostly
OK but you could lose the odd one. Something a skilled gardener could
make an educated guess on if they are able to see the plant in the flesh
rather than in a photo.

Rod

I've just thought that another avenue that may be open to you would be
propagation of 'new' bushes from your 'old' rose bushes. Taking cuttings
should be an option, but success will depend on the type of roses you have.
Look in the RHS book on propagation or in any good book on propagation and you
should get some good advice. I'm not sure whether layering would work as
well/instead.

All the best.

Dave.