As you suggest, Kloofneck, you can't tax the new vehicle without getting it insured first. You won't be able to use the online procedure because that's only available when you're the registered keeper of the vehicle (and it takes some time for the DVLA records to be updated).
So you'll have to go to a Post Office. You don't need to get an insurance certificate posted to you first though. If you insure via a local broker they can issue you with a cover note (which is acceptable at Post Office counters). If you insure online you'll be able to download a certificate of insurance. (Downloaded certificates are also valid at the Post Office).
Pleasae ignore Uriah Heep's post above. There are two entirely separate pieces of legislation. One says that it's illegal to drive a car without a valid MoT certificate, while the other says that it's illegal to drive an unroadworthy vehicle. Failing an MoT test doesn't invalidate an earlier MoT certificate. You can continue to drive a vehicle, after its failed an MoT test, using the 'old' MoT certificate certificate unless the reason for failure was one that declared the vehicle to be unroadworthy. (e.g. a vehicle could fail an MoT because of badly corroded brake pipes, which might well not last another year, but still be roadworthy at the time and therefore legal to drive).