You don't need ILR to get your license. In fact, you need to apply for your license now as US license holders can only drive legally for 12 months here. You'll need to send your passport, a photo, the application form, and the fee to Swansea and you'll have your provisional license in about 2 weeks, which will only allow you to drive when accompanied by a UK license holder over 21 with 3 years of driving experience. From there, you'll need to take your theory test and then your practical. Once you pass both of those, you'll apply for your full license.