In my opinion, it shouldn't make a difference. Personally, I store dates as UNIX timestamps too, and if you're using a method like:
getSecondsSinceEpoch = ((date) => Math.floor(date.getTime()/1000));
(or any other function that utilizes Date.prototype.getTime()), then you are storing a universal date that is independent of timezones. This can be beneficial because regardless of the user's timezone when they log in, you will not be returning values based on where they were previously located. The same principle applies when Daylight Savings Time alters the clock in their region (or remains constant after being abolished).
To display the correct human-readable date string for the user's specific location, Javascript uses a timezoneOffset, which determines how many time zones away the browser is from London at that moment (in minutes instead of hours, and adjusted for Daylight Savings Time). Presumably, the user's device and browser should update accordingly to reflect this change, ensuring your script behaves correctly. (If the user's device has the wrong timezone set, it may lead to issues not just with your app but with others until resolved).
While I am not an expert in this area, based on my current knowledge, I believe your approach should work fine. If there are inaccuracies in my explanation, I hope someone more knowledgeable will provide clarification soon.