There is some history of this problem in the forum going back a number of years. A fair bit of discussion but no definitive answer. About four months ago my otherwise reliable Omega began to suffer with this. On most start-ups the rev counter and speedo worked fine. Stop the car and switch off after a few miles then restart resulted in no rev counter or speedo function. This habit continued for a month or so then both instruments decided to cease working at all. This did not bode well as an otherwise perfectly good - and lovely driving - car was scrapyard bound as no speedo is illegal to say the least. Shortly after this total failure and when taking her indoors to a hospital appointment, the old girl (ie the Omega, not her indoors) failed to start. Battery quite knackered as only got about half a rev off the starter. Odd how modern batteries are apparently quite fine one day then fall off a cliff and die with no indication. Anyway off to local tyre and battery place for a new one. Fitted without problem. Started OK and to my astonishment, both rev counter and speedo were fine, working perfectly. This has been the case for the past few months. From this I draw the following conclusions:
1) Pure coincidence. Whatever the fault was has rectified itself.
2) Disconnecting and reconnecting the battery has prompted some sort of 'system reset' such that the instruments function OK.
3) Fitting a fully-charged battery provides full-bodied voltage that perhaps was lacking with the old and dying one. The instrument systems then 'woke up' with this higher voltage and all was OK.
Personally I favour option (3) but I leave to readers to make up their own minds. In any event, if your Omega is suffering this problem, the above is a relatively quick test. It just might work...