"You need a power supply voltage of at least 13.8V to charge a 12V lead acid battery."
13.8V will give you no charging current.
Wow, you mean for the last 40 years I've been using ~13.8V as a standard to see if my customers' 12V Lead-acid batteries would charge after I've left the premises? that I've actually been leaving them with a system that wouldn't work? Amazing that all those thousands of customers didn't complain about my service sooner or later! Also amazing that their batteries continued to work through all the occasional power outages year after year without the batteries dying from lack of being re-charged from a charging circuit that topped out around 13.8 VDC.
All sarcasm aside, a lead-acid cell that is fully charged will read ~2.2VDC, OCV (Open Circuit Voltage, i.e.. no load connected); so a 6-cell battery will read ~13.2V when fully charged. The "12V" is a "nominal" designation for a battery that seriously needs charging if it actually reads 12V, OCV.
For an accurate OCV reading to assess charge, the battery needs to lie inactive (no charging or discharging) for a couple of days, but you can get an approximate reading after a few hours. Taking an OCV reading right after it's disconnected from charger or load won't give an accurate reading.
13.8VDC
will push a charging current through a lead-acid battery, even if it's fully charged already--which is why you need some kind of regulation on chargers: So they don't overcharge batteries.
I'm curious, Colin: Why did you think it wouldn't?
EDIT: @Pidja, some power supplies and batteries actually operate at a slightly higher voltage than their ratings indicate. If your "12V" (I assume it's actually "12VDC"? ) power supply actually has a constant 12VDC output, it's not high enough to charge your "12V" lead-acid battery.
If you have a digital multimeter (DMM), check the output of the supply. If you don't have a DMM, I recommend you not try any DIY electronics before you acquire one.