Unless it's an emergency or a condition where there are deadlines for treatment, eight weeks is not long for NHS appointments. There are criteria for treatment times, eight weeks may well be in the acceptable band for your wife's condition.
Changing GPs will only complicate the matter. You need a straight answer from the consultant as to when they wrote to the GP. You can then go to the GP practice saying you know that X wrote on x date asking for a referral, please, can they tell you when the referral was made for the scans. They ought to be able to do the ECG in the surgery now, so I don't know what the delay is there. I have to say that at this time of the financial year, many health areas are running out of funds (as per usual) so will be prioritising urgent cases.
I don't understand the answer you're getting about "there isn't a problem" - all you want to know is where your wife is, in the queue.
BTW - no point in losing your temper about it, just keep checking with the GP who is requesting the appointments for you. The private consultant won't know. it's nothing to do with them any more, you've gone back to the public sector for your wife's next investigations.