A very important word of caution for restoring, repairing or dismantling old compasses.  The luminous paint used up until the 1960's was based on Radium, which is highly radioactive and VERY DANGEROUS.  The Radium will remain lethal for up to fifteen thousand years and must not be ingested or inhaled as fine dust particles.  Protection and optimum safe working practices are essential, and this work should only be carried out by trained and qualified personnel.  Current U.K. compass manufacturers simply refuse to handle compasses containing Radium due to the risks involved.

Another risk to consider is that green compass cards from the 1800s and their respective makers’ green labels stuck inside the boxes were dyed with arsenic, and are hazardous even to touch, as arsenic can enter the blood stream through the skin, as well as through inhalation or ingestion.  Arsenic poisoning can have similar symptoms to cholera.

I cannot overstate the dangers of dismantling these old instruments.  If your compass has thick brown, red, or pinkish marks on the dial then it can be safely assumed it contains Radium and should not be dismantled.
A Geiger counter is advisable if you intend to collect old compasses to give an indication as to which you can dismantle for restoration, and which should be left alone.  It is important to understand that the danger is less in the actual radiation than in the inhalation or ingestion of contaminated dust or compass fluid.

Below is a photograph of a Geiger counter reading on a 1915 British compass and you can see 9.99 micro Sieverts per hour.  This reading would actually be very much higher if the counter didn't scale out at 9.99, so to look at this in perspective and to be correct, if you were to keep this compass in your pocket you would receive the maximum permitted annual dose in 20 / 0.0099 = 2030 hour (or about 83 days).  In reality it’s a bit more complicated as the 20 milli Sv limit is for “whole body exposure” and different (higher) limits may apply to any localised exposure.  The REAL danger, however, is ingestion, inhalation or bodily ingress through the mucous membranes of radioactive particles released into the air when the old paint or fluid is disturbed.
Maximum permitted dosage for personnel working in the nuclear industry is 20 milli Sieverts per annum.  Treat these items with respect.  More detailed information can be found in the Internet from many official sources.
The UK Health and Safety Executive states:

The practice of luminising the hands and dials of clocks and watches dates from early in the twentieth century and traditionally used the radioactive material Radium-226.  This material emits both alpha and gamma radiation and as a result has a considerable "radiotoxicity" associated with it.  The radiation risks from work with luminised materials can broadly be divided into those from internal radiation that results from radioactive materials becoming absorbed into a person's body, and those from external radiations from persons being in the vicinity of or handling radioactive materials.
Below right is a compass from 1940 clearly showing the brown / pink paint typical of Radium-based luminous paint used in the past.  This compass, like the compass below-left, drives the Geiger counter right off the scale.

All the Radium in this compass is safely behind glass and as long as the compass remains intact there is little chance of inhaling or ingesting Radium dust.  It is, however, advisable to not keep such items within five metres of where you spend long periods of time, such as near where you eat, work, sleep or watch television.
The danger can also come from incorrect handling of breakages.  In the event that you should drop such an instrument and the glass breaks, DO NOT USE A VACUUM CLEANER to clean up the pieces.  This will suck in the Radium powder and blow it out into the room for you to breathe.  Instead, spray a fine spray of water everywhere and using gloves and a protective face mast clean the area with damp cloths to ensure you collect all the dust and then safely dispose of the gloves, mask and cloths.
Remember also that any liquid inside the compass will also contain and emit substantial quantities of Radon gas due to contact with the Radium for many decades, and should thus be treated as if it were actually Radium.  Radon has a half-life of around 18 hours, so if the fluid is left open in a safe, well ventilated place, for about 5 days it will then be safe to dispose of as non-radioactive waste.  This is not the case for the actual markers, they will remain highly radioactive for many, many thousands of years, even though they no longer glow in the dark.
The best advice, however, is that if you suspect your compass or old watch has Radium paint, even though it has lost its luminescence, DO NOT REPAIR, RESTORE OR GENERALLY TAMPER WITH IT.  Just treat it with great respect, handle it occasionally, and keep it away from where you spend long periods of time.  If you really do have to repair or restore, then seek expert and qualified assistance.