As a first notable update, mostly of interest for @alberto: we looked for existing cash flow prediction software, to not reinvent the wheel if we don't have to. However, the results were disappointing: no open source alternative is available at all, and the commercial ones seem too expensive for our needs. (I'm personally a big fan of keeping business running costs low, and paying more for a small financial software than we pay for all of our Discourse hosting would be certainly strange.)
Here is an overview of all serious candidates we found, in case somebody wants to argue that we should use one of them instead of rolling our own:
And some options which, against our expectations, can not provide a suitable solution:
QuickBooks Online: it does not support cash flow predictions (see); note that QuickBooks Desktop does, but we need an online solution
Easy ERP: an open source ERP solution that comes with a cash flow feature, but that is seemingly only about past cash flow; other open source ERP software might provide forecasts, but also a ton of features that we don't want to have, so it would be worse than a simple, tidy spreadsheet
Open source personal finance tools: Some of these tools, esp. seemingly Skrooge with its "Simulations", offer cash flow predictions. However, they are all desktop software, so not suitable for a simple collaboration collaboration process. Moreover, they are not made for business purposes, so might not have features like per-project budgeting etc. that we need.
Our chosen path is so far to develop a spreadsheet template according to our own needs. We found quite some examples to look at, and even our accounting software FreeAgent recommends this approach. So this seems to be the usual path taken by businesses which are too small for a commercial cash flow tool or ERP solution.