What would be the consequence of using Energy Saving Bulbs in these
table lamps?

It depends on the timing device. These usually work in two different ways.

The more expensive types and those with higher ratings use an
electromechanical switch called a relay. These can switch most types of
load, although whilst switching slightly reactive loads such as energy
saving bulbs, the contact ratings are lower. This is rarely a problem, as
lighting is low power and the contacts are usually rated for 13A to control
heaters and the like. This design requires a neutral connection. And gets
power between the unswitched live input and the neutral.

The other type of switch doesn't require a neutral connection. They run
entirely inline on the live connection. This enables them to be a direct
replacement for a light switch where no neutral is available. When they want
to switch on, they go almost a complete dead short, and extract the energy
required to run by having a tiny voltage drop. When off, they get energy by
simply taking it between the two terminals. This means that a tiny current
will flow through the load. As this is usually a filament lightbulb, this is
no problem and the current is too tiny to be noticed. However, an energy
saving bulb is a discharge tube that will not pass any current until a high
voltage is presented. This means that the switch doesn't get the tiny flow
of current required to power the electronics and will not work. Sometimes,
the problem is even worse and the small current passed by the electronics
builds up voltage in the bulb's starting capacitor meaning that the bulb
will flash every minute or so. This depends on the design of the bulb's
electronics.

So, whether it will work or not depends entirely upon the design of the
timer switch.

Christian.