the law is clear
its illegal to drive without valid road tax (Vehicle Excise Duty)
its illegal to drive without displaying a valid VED disc
I think you are going to struggle to get the car insured until you have the V5 Vehicle registration in your name, not just an insurance certificate. the insurers shoudl send a cover note out by first class post, so theoretically it should be with you today or possibly tomorrow
what you could try is seeing if you can insure it online on the DVLA website. theoretically the insurer should have informed DVLA of the insurance, you should haev registered the vehicle in your name.
the problem at present is that although you have insurance it may not have been updated on the police database, so you are possibly more likely of gettign stopped by ANPR equipped vehicles as having both no tax and no insurance.
so driving (or even parking) on the road is breaking the law. if you drive it you are taking a risk that you wont get stopped, or if you do the police will let you off. they may let you off if you come across as law abiding, helpful, friendly and so on. if you have attitude then they will not exercise what ever discretion they have. so ultimately its up to you. if you don't think you are going to drive by any ANPR equipped cars, cameras or wahtever, and you don't think you are going to get stopped by foot patrols, and don't expect to park in a public palce then you may consider taking the risk. personally I wouldn't. if you do get stopped its probably a fixed penalty that needs reporting to your insurers who will no doubt with great glee rack up the insurance fee. there is another risk which is you have just passed your test you have got 3 points on your licence in your probationary period. I forget the details but if you get (IIRC 6 or more points in the first 12? months the courts can revoke your licence meaning you start as a learner all over again