Does it cost me anything extra to use my Apple Watch without my phone nearby?

What happens is you pay an extra monthly fee on your existing phone contract, the watch uses your existing phone number to make calls.In the US it's about $10 per month and about £10 in the UK,that is for the cellular version of the watch, if you buy the GPS only version of the watch you can make calls and send texts as long as you have your phone with you,with the cellular version of the watch you can make calls and send texts without having your phone with you.So in reality your paying $10/£10 a month so that you can leave your phone at home.

No. If you have a GPS+Cellular watch you are already paying a monthly fee (in addition to your iPhone plan) to use the watch without you iPhone. If you have a GPS only version, it doesn't matter anyway since it doesn't use cellular data..