The original alloys that came with the car, although I like them a lot as they have the original style and feel of the original car, are actually very annoying in as far a they are metric. This means tyres are a problem.
The tyres are 255/55/R390 and its the R390 which is the give away that it is a metric tyre. Metric as in metric radial dimension as opposed to the more commonly used imperial measurement in inches, e.g. R15", R17" etc. For my tyres, 390 / 25.4 = 15.35". I believe a lot of E34s were later sold with imperial wheels of the same style, so perhaps I just got unlucky.
The best solution the garage suggested was to get part worns; find them second hand or from a scrap yard or better still get rid of the wheels. So this is what I did. I kept my eye on ebay for a while and finally found a suitable set of 4 wheels with still legal tyres.
The new wheels are from an E39 so I am happy to use these as they are at least original BMW 5 series wheels and I am very happy with them. I don't think they look out of place too much on the car, I didn't want to end up with something that looked wrong on the classic look of the E34 Touring. And of course the best bit is that they are 225/55/R16... a standard size that will allow me to buy normally priced tyres from anywhere.
Even though the front control arms on the car are badly worn and affecting handling quite a lot, the old wheels and badly worn tyres were obviously having a negative effect too. The handling is noticeably better just with decent tyres/wheels. There is not as much vibration at speed on the motorway and low speed rhythmic groaning I was getting (sounding like a worn wheel bearing) has gone.