My bike has been subject to two non-fault claims:
1st one was 10 days after I bought it - decked by a works van in the company car park (Tuesday). Put bike into my usual bike shop 2 streets away, got an estimate (£1246 IIRC all up, parts / labour / VAT) and put it into my employers. Employers insurance (Aviva) came back on the Friday accepting liability, and had the bike assessed by the following Friday. Bike was back home within 2 weeks of being assessed.
2nd one was also in works car park, decked by a workmates car. Went through same procedure of getting the estimate, and mate gave me his insurance details (Axa). Axa were slow to respond, so mate contacted them admitting liability. Axa came back wanting bike to go to their "approved" repairer. An email request to use my normal shop was approved after supplying estimate details, the bike were assessed, and the work was done. Again, I had it back a couple of weeks after that.
They don't need to obtain parts from Japan anyway - most garages will obtain parts from Yamaha or Fowlers.
One "trick" I used in a (non-fault) car claim was to tell them I *absolutely* needed a car (I occasionally do contract work for a specialist engineering firm) ie "I have to go to a site in Newcastle (true), they're going to stop work for 4 days, losing £100,000 per day, so I can do the job (also true), if I can't get there because I'm without a car the job won't get done. If it isn't done and they want their £400,000 reimbursed, I'm going to put them in your direction". "Oh, we can't pay that" says 3rd party insurer. "Your customer caused the problem, so you best get something sorted" - it got sorted VERY quickly after that.