Re. KC and UKA measuring - yes, you do have to be measured for both. It's because UKA isn't directly affiliated to the KC. KC measuring needs to be done before you enter shows. KC official measurers do measuring sessions periodically, sometimes at shows, sometimes elsewhere. I think you'll find them advertised on agilitynet. With UKA it's more informal as they seem to hold measuring sessions at most shows, so you could actually get measured at your first show.
UKA Nursery Agility classes don't have weaves, seesaw or tyres, so that would probably be your best bet at the moment.