Even if the car was totally unroadworthy the legally-required third party element of your insurance couldn't be invalidated. (i.e. you couldn't be prosecuted for having no insurance, as the insurer would still have to pay out to a third party if you caused an accident. It's possible though that they might then seek to recover their losses from you if you'd failed to keep your car in good condition AND that failure had actually resulted in the accident occurring).
Additional cover (such as that in a 'fully comp' policy) is subject to the specific terms contained within the policy but it would be hard to see how simply not having a functioning odometer could affect your cover.
Not having an odometer isn't going to increase the likelihood of you having an accident, so it's unlikely to influence your insurance cover. Not having malfunction indicator lights would probably not be relevant either UNLESS, through not having a relevant warning available to you (about, say, the loss of power steering fluid) you lost full control of the vehicle, resulting in an accident.