Lawn best mixure for UK's Caly Soil

Hi,

First of all I wish to thank for every user of this forum that helped me to resolve lase issue with soil depth and soil preparation.

I removed rubble from my backyard, used Garden Lime (to neutralize Ph because my Hydrangeas were pink ...btw I highly recommend them instead of testers) and added 120l of Soil Conditioner for 30sq/m area with slightly mix with top of my soil (quite smelly :D).

Next I used pitchfork to make 15 - 20cm holes in whole area (a lot of them) and I put aprox. 50kg of dry garden soil into those holes. Rest of garden soil in on the surface (max 2-3mm).
In some areas water still don't sink well (sink after 1-2h at night after heavy watering). So I will mix some hard garden sand (2-4mm stones) with clay there to improve aeration there (I have some of it after digging).


Now the question ...

Which Lawn Mixture will be that best for Clay Soil (after above preparation) in West Yorkshire UK area (quite rainy most of the year) for usage like Recreation, Barbecue with no kids or dog or any sport activity and with quite fast germination time (I don't mind to spend some extra time on mowing but I don't want to wait 2 years for lawn as I live in rented house).


Please include some real life mixtures that I can buy in Garden Store or Online (incl. brand etc.)


Some of 'wise' internet tutorials says that Bermuda Grass will do the work with nice 'carpet like' lawn and It probably doing well in clay soils. But naturally it grows in sandy surface so I am not too sure about it.


Best Regards
Michael