How can I update my HTC Desire when the OTA update is missing? I have an HTC desire and have Orange as a provider in the UK. The software information displays my firmware version as being "2.1-update1". 

I discovered that my phone is not up to date, because after hours of searching I discovered that the reason I cannot find the google app 'phone to chrome' is because the version is not recent enough. There are no updates pending as I have the system software updates check box for schedules updates checked. On checking for updates a message states that there are no updates available for my phone. Is the reason that other users are able to update because they have different devices and service providers? From looking around for update news, my phone should be updating.

From various webpages (example) it is said that this update can be done manually by using HTC sync; is this safe? Are there risks from not getting the update automatically? Or are you only bypassing the service provider in this way and not much harm can be done? A warning that this removes all previous saved data is given, but this will not equate to rooting and voiding any warranties though?
Yes, it's up to the carrier whether to distribute the update over their network, but it looks like the update did roll out on Orange.  Strange that you didn't get it and can't see it.

That said, updating via HTC Sync is perfectly safe and will not void your warranty.  It's nothing like rooting.

I would think you should be able to take it into an Orange location and they'll update it for you, but I would just do it with HTC Sync myself.  Your link is for the 2.3 update, rather than the 2.2 update; I don't know whether you can go directly from 2.1 to 2.3 with that update, but it shouldn't hurt to try.