I could imagine that interest rates might possibly look better in a few years, but I wouldn’t bet on it. As far as construction costs and generally building material shortages and things like that go, the situation isn’t the easiest right now, but I also don’t see a huge turnaround in the near future. So it already seems today that building material shortages are shifting a bit (e.g. wood is more relaxed again, roof tiles are scarcer), possibly it will again ease a bit in the next few years, so there might possibly be potential for building materials to get cheaper... on the other hand, it could very well be that some issues will occupy us for more than 5 years (which is too long for you) and even then, you might be better off today because general inflation on other materials and wage increases might offset the price decreases from currently overpriced materials. Actually, I see really cheaper new-build houses more through cutting wishes and focusing again more on the essentials and much less through currently inflated prices. (E.g. there were several new-build townhouse projects here in a B-city for 850k+ with a lot of frills... terrace + 2 balconies, smart home, garage etc pp – a year ago that was more within some people’s affordable range than today... even higher earners are cutting back on a lot of frills right now)
But that is something you can already address today for your project as well: reduce to the essentials, possibly identify a few spots where “this can be added later.”
My source here is only my crystal ball and my conclusion: no idea! If you can afford it today, I’d probably start potentially today as well. Ultimately you’d also get out of the rental situation and start paying off. If you can’t afford it today... well, then you can either see if you can downsize enough to make it affordable or have another cup of tea and see what happens? If it’s affordable, then in the worst case you just didn’t make the best deal.
For myself, I have also realized this: Of course I would be annoyed if interest rates massively drop next year and I’m still stuck in the credit contract with high rates for 9 more years (we have a longer fixed interest period, but after that you can cancel). But as long as we don’t have to turn every cent five times, but can still live quite comfortably, it’s ultimately not that important to me.