What I've forgotten to say: I want to program it via the UART-Interface...
Here's some background that will help you along in this process as we do not provide any step by step instructions in this area, but many have accomplished the same using the information here on the community.
First off, the part MUST be programmed the first time using the HCI UART. This UART MUST come up and be available within the "programming mode' (lots of threads on this subject). Most customers create some type of header on their board that can accomodate the HCI UART (Rx, Tx, PWR and GND), along with a means of assuring that the part either comes up in programming mode or can accomodate being manually placed into programming mode.
Once you have the above HW changes made, you will then need to understand that in order to program the device through the HCI UART, a proprietary protocol is used which is lincluding within the "loader" executable that is part of the SDK and the command line tools described here: Programming the TAG2/TAG3 Board using command line tools
In this thread I go into a great level of detail as to how this 'programming' process takes place at a SW level: Re: Production Programming of BCM20737S
Now in order to attach the PC running the SDK/Command Line Tools to your board with the designated header, you will need some form of USB to serial programming cable. For this, we highly recommend using the FTDI 3.3V cable jcardona describes here: Re: BCM20736S recovery problem - SDA high does not enter recovery mode
These two threads also discuss this process: