It is vital that the person doing the job understands the critical dimensions and has a system for ensuring their accuracy. Otherwise handling problems will arise and in extreme cases may be dangerous. There are many stories of conversions going wrong and mostly they relate to positioning of critical steering components. Fixing mistakes in this area after the car is completed can be extremely expensive and involve major rework. Often a conversion service will have greater strength in one of these two areas.
Reputations come and go - usually with the people.
Speak to the person doing the work and make sure they are happy for you to be involved. When the word gets out, someone good may become very busy and the wait may be many months. They may also employ others and quality may suffer. It is probably best to find someone who has had a similar car converted to a standard you are happy with and ask for the same person to do your work. Be assured conversion to right hand drive is a huge job.