According to Google/Gemini 1976 had 5 days at or over 35C and this is still so far the record

https://en.wikipedia.org/wiki/1976_British_Isles_heatwave 

It would be really useful to have a detailed dataset from (say) 1850 of all the data shown in Met Office "24 hour extremes in the UK"  available for analysis in one file

I don't think such a thing exists or even if the older data is even recorded, It would be a huge task to compile lit from the Met Office daily archives but it would answer a ton of the queries we see on here 

Yes AI said six for 2022, but I find AI quite poor when it comes to UK weather statistics.  Definitely some sort of database would be useful 

Unless I am missing something, I don't think the older Met Office summaries (pre 1990s) even include an overall max for each day, they just seem to show the maxima for the various stations, so it might not be very easy at all.

Roost Weather has a good feature where you can get the summary map with the top 10 temps for any given day right back to the 19th century (although records are obviously more sparse back then). Unfortunately though if you try and look at too many days at once it blocks your IP, presumably to prevent data scraping.

https://www.roostweather.com/ukobs/maxtemp_map.php 

From what I can see, 1976 and 2020 both had 5 35C+ days. 1990 and 2003 both had 3.

2022 only had two that I can see: July 18 and 19. August 13 came close with 34.9.