Sorry for your loss.
AFPS75 is a Government Service pension in terms of the double taxation treaty, so taxable only in the UK and subject to Progressionsvorbehalt unless you have German citizenship, in which case it is taxable only in Germany and does not feature at all on your UK tax return. If the latter applies you will need to send a form DT Individual (Germany) to HMRC to enable the pension to be paid gross by AFPS, with no tax deducted.
If it remains taxable only in the UK it will be subject to PAYE to collect the tax and if you wish, and if the rental income is less that £12,570, you should be able to have that "coded out", i.e. set against your personal allowance so that something approaching the correct amount of tax on pension and rental income should be collected via PAYE by the end of the year.
e.g. if you do nothing in this regard, your widow's pension would get a PAYE tax code of 1257L, meaning the first £12,570 of that pension is not taxed and any pension above that is subject to PAYE at 20% (assuming that pension is less than £50,270 overall). Your UK tax return would then show the gross pension, tax collected under PAYE and your net profit from your rental. Additional tax would be shown as due in the tax return based on that rental profit. If you decide to have the rental income coded out, and if it's, say, £5,000, your tax code would reduce from 1257L to 757L, meaning that an additional £5,000 of your widow's pension would be taxed at 20%. Your tax return would look the same, except for the entry for the tax already paid, which would be £5,000 x 20% more because of the rental income being taxed via PAYE and meaning less tax to pay at the year-end. At the end of the day, the tax payable is the same, it is just spread over the year if collected via PAYE.
I'll leave others to comment on the KK position in Germany.
You may find this article about the AFPS75 widow's pension worth reading
https://forcespensionsociety.org/2021/0 ... explained/