It does say in the paper form that they will calculate the tax due. (form SA100 Page TR6). I also had the option of calculating the tax due myself. I chose to let them calculate it for me, based on the information I provided to them. I am actually not expecting a tax bill this year, as the only income I reported that should be taxable in the UK is about 8K worth of 403B distributions (but those have already been taxed in the USA, which I did note). Next year I will potentially be paying UK taxes on my Social Security income that goes over the personal allowance.
So, eventually I will see the updated info online and get a paper response from them. It took a week for them to officially receive and log my paperwork, so that means it's only really been about three and a half weeks since I sent it in. I will start checking online in a few more weeks, then.
Thanks.