OK here are some stats.
It is really quite difficult to get directly comparable data. I have not been able to find all the information I want for both countries in the same year, but this is the best I can do. Some of the most detailed data for France is taken from an interactive map of all the maternity units in the greater Paris area which compares the statistics for that unit against the national average, but they are compared against the national average for that type of maternity unit, ranging from a basic level 1 maternity unit to a full service level 3 maternity unit with an attached NICU. So when I give a range, that is why.
France
C-sections:
19.9-21.6% in 2021 depending on type of maternity unit according to the interactive map or 21.4% overall according to the 2021 perinatal health enquiry.
Information from the French health ministry clarifies that in 2020 the overall C-section rate was 20.4% of which 13.9% were EMCS and 6.5% were ELCS. This is expressed in the link below as 68% and 32% of C-sections respectively.
The most common indication for a C-section is having had a previous C-section birth, which ties in with my own experiences. The VBAC rate appears to be significantly lower in France than in the UK.
Spontaneous vaginal births:
64.2-68.3% in 2021 depending on type of maternity unit according to the interactive map (significantly lower for first time mothers in all cases). It's not entirely clear whether this means non-induced vaginal birth or non-induced, non-instrumental vaginal birth. Unfortunately the interactive map doesn't give induction rates so I've had to take that from a different source.
Inductions:
25.8% in 2021 (has increased from 22% in 2016) according to the 2021 perinatal health enquiry.
Epidurals:
77.1-83.7% of vaginal births in 2021 depending on type of maternity unit according to the interactive map or 82.7% overall according to the 2021 perinatal health inquiry (decreasing as more women opt for physiological methods).
Assisted vaginal births:
11.6-13.6% in 2021 depending on type of maternity unit according to the interactive map.
Note: It's not clear from this source whether this is a percentage of vaginal births or a percentage of all births.
Episiotomies:
2.9-3.8% in 2021 depending on type of maternity unit according to the interactive map.
UK:
I found it much harder to access the same statistics for the UK and it's not clear whether the data used relates to births which actually took place in 2021 or whether there is a time lag. I've done the best I can.
C-sections:
27% in 2021 (16% EMCS and 13% ELCS) according to NHS Digital.
Spontaneous vaginal births:
53% in 2021 according to NHS Digital. They clarify that this means non-induced, non-instrumental vaginal births.
Inductions:
33% in 2021 according to Summary Report 3 of the Excel data set for 2021 used by NHS Digital (you need to scroll to the bottom for this).
Anaesthetic/analgesic use:
59% in 2021 according to Summary Report 4 of the Excel data set for 2021 used by NHS Digital.
Note: I am not actually sure what this means. Obviously it would include all vaginal births under epidural, but I am not sure whether it includes C-section births or the use of drugs such as pethidine.
Assisted vaginal births:
According to NHS Digital and the related dataset the rate of instrumental births in 2021 was 10% overall, which is slightly lower than France. The National Maternity and Perinatal Audit (published in 2022 but using data from 2019) puts the rate of instrumental births at 23% for a first time mother and 12.3% overall (you need to click on the full report) which is about the same as France.
The NHS Digital page clearly expresses the 10% as a percentage of all births, not as a percentage of vaginal births.
Episiotomies:
The only data I could find about episiotomies was from the National Maternity and Perinatal Audit which says that 44% of the first time mothers who had a vaginal birth had an episiotomy. That seems shockingly high to me, especially since the data comes from a time when reading Mumsnet put the fear of God into me about episiotomies being performed routinely in France.
Anyway, that's the best I can do. I really hope these links work.