1 min googling, top result:
US suicide rate
In 2018, there were 48,344 recorded suicides, up from 42,773 in 2014, according to the CDC's National Center for Health Statistics (NCHS). On average, adjusted for age, the annual U.S. suicide rate increased 24% between 1999 and 2014, from 10.5 to 13.0 suicides per 100,000 people, the highest rate recorded in 28 years.
England suicide rate
The suicide rate of 11.2 deaths per 100,000 population recorded by the Office for National Statistics (ONS) in 2018 is an increase on the 10.1 per 100,000 population recorded in 2017, which was the lowest since the organisation began recording data on suicide in the United Kingdom in 1981.
Singapore suicide rate (considered the strictest on guns)
There were 397 reported suicides in 2018, compared with 361 the year before. This raised the suicide rate to 8.36 deaths per 100,000 Singapore residents, up from 7.74 in 2017. All age groups, except those aged 60 years and above, recorded an increase in the number of suicides, said the suicide prevention agency.
These are overall suicides, not by method. I highly doubt there's any statistical significance in proving that no access to guns results in people not wanting to kill themselves. The argument literally then turns to, "Because guns exist, people want to kill themselves. It has nothing to do with relationship, ptsd, financial or existential issues. Strictly their access to guns."
That uptick the US had for suicides, how much you want to bet that's due to neglected veterans after Iraq and Afghanistan? Looking at the bottom chart for US suicides: https://en.wikipedia.org/wiki/Suicide_in_the_United_States#:....
You have a general increase of suicides throughout the Vietnam war and then for about 3 years after it ended (1975). In the little bit of googling, I can't find solid numbers, but during that 2000s increase timeframe, there was also a report that more vets died by suicide than by actual conflict in Iraq and Afghanistan.