Just to recap and to make sure we are all on the same page.
You said you bought new wheels, I've took this to mean we are talking aftermarket ones here not a recon set of originals as suggested above, is this correct? You've already proved the issue looks like a wheel balance issue on the front wheels as it vanishes when the wheels are swapped front to rear. that just leaves the wheels and the tyres as possibilities.
Here's a few things to check before throwing any more money at it. Have a look at the front wheels and count how much weight has been added, it will give you a good indication of how good the wheel/tyres are. Lots of weights usually points to few things;
Poorly manufactured wheels.
Cheap tyres- these often need more weights than premium brand equivalents
Badly/unevenly worn tyres.
Beyond the wheel weights themselves this can also be due to a slightly out of shape wheel or tyre. I seen this first hand with cheap aftermarket wheels and cheap tyres. In fact in the case I saw with 4 brand new cheap tyres the car also made a right old din like the wheel bearings had gone.
As I've said it can only be either one or both of the wheels or the tyres. I know you trust the place you have used, but the thing to do now is go to another reputable tyre fitter and see if the balancing is actual as it should be, this eliminates a dodgy balancing machine from the equation. I would also be looking at the wheels and make sure the inside surface where the bolts go through/meets the hub has not been painted. On both aftermarket and recon original wheels painting this face can cause exactly the issues your having.
You said you bought new wheels, I've took this to mean we are talking aftermarket ones here not a recon set of originals as suggested above, is this correct? You've already proved the issue looks like a wheel balance issue on the front wheels as it vanishes when the wheels are swapped front to rear. that just leaves the wheels and the tyres as possibilities.
Here's a few things to check before throwing any more money at it. Have a look at the front wheels and count how much weight has been added, it will give you a good indication of how good the wheel/tyres are. Lots of weights usually points to few things;
Poorly manufactured wheels.
Cheap tyres- these often need more weights than premium brand equivalents
Badly/unevenly worn tyres.
Beyond the wheel weights themselves this can also be due to a slightly out of shape wheel or tyre. I seen this first hand with cheap aftermarket wheels and cheap tyres. In fact in the case I saw with 4 brand new cheap tyres the car also made a right old din like the wheel bearings had gone.
As I've said it can only be either one or both of the wheels or the tyres. I know you trust the place you have used, but the thing to do now is go to another reputable tyre fitter and see if the balancing is actual as it should be, this eliminates a dodgy balancing machine from the equation. I would also be looking at the wheels and make sure the inside surface where the bolts go through/meets the hub has not been painted. On both aftermarket and recon original wheels painting this face can cause exactly the issues your having.

, I had the tyres fitted mobile while I was at work however when I spoke to the tech a moment ago he said the wheels balanced perfectly and one of them didn't even need any weights