When you receive your Tynemouth Bicycle Company basket the straps will be positioned close to the right and left edges of the basket which is roughly where they need to be to fit North Road / "Sit-Up-And-Beg" type bars (also sometimes called comfort handlebars). In this first picture showing my wifes bike you can see that all I've had to do is strap the basket on and its ready to go. With some bikes you will need to slightly adjust the position of the brake levers, more on that below.
In this second picture you can see a basket fitted to a set of mountain bike type riser bars, these are the original bars that came on my wife's trek hybrid bike and are now on an old mountain bike of mine. As you can see in the photo I've needed to move the straps inwards towards the centre of the basket. To do this just gently pull the strap out and then thread it through the wicker. If you can problems re-threading the straps you can push the strands apart a bit with the end of a spoon.
For this basket which I fitted to a neighbours Giant bike I've had to move the straps even further in, theres no problem in doing this. I also had to adjust the brake levers and slightly change the angle of the handlebars which I'll explain how to do below. In fitting the basket it is important to make sure that the cables are not compressed in such a way that it prevents brakes and gears from working correctly (check them after installing).
There is no problem with a basket being in contact with the brake levers as long as it does not interfere with the working of the brakes. The levers can even help keep the basket at a good angle, but sometimes they can prevent you getting the basket close enough to the bar to strap on.
Underneath each brake lever there will be a allen bolt which clamps the lever on to the handlebars. If you loosen this off by a couple of turns you will find that you are able to change the angle of the brake level up and down. The angle of the brake to the handlebar should be roughly 45 degrees but you can vary either side of this to make it easier to fit the basket. Similarly the handlebar itself will be help in place by a clamp at the top of the stem secured by two or four allen bolts, if you loosen these a little you will be able to change the angle of the bar. If I'm fitting a basket to a fairly flat barred bike I'll loosen both the handle bars and the levers before I start so I can move them around while I'm fitting, and then gradually tighten up as I get a feel for where they need to be in relation to the basket.
WHEN YOU ARE FINISHED ALWAYS TIGHTEN AND THEN RETIGHTEN ALL OF THE HANDLEBAR AND BRAKE FITTINGS
CHECK THAT THE BRAKES AND GEARS WORK PROPERLY BEFORE YOU RIDE
GET YOUR BIKE SERVICED AT LEAST ONCE A YEAR BY A GOOD BIKE SHOP IF YOU RIDE IT REGULARLY
If you are having real problems getting it fitted feel free to contact me with some photos and I'll do my best to help. If you like the idea of having more traditional type handle bars then a set of Raleigh North Road handlebars can be had for around £15 on ebay or through your local bike shop and its a then minute job to transfer over the levers and grips to a different set of bars.