I did a test before travelling to London recently. The nearest SuC was 24 miles away and I started with a cold battery but despite pre-heating I only saw 24kw on the couple of stalls available. Not shared. So I gave up and came back. During the actual trip at Reading (16 stall) I saw ~60kw and only on the way back did I see the 100kw max (at the time) on the way back at the Membury West service station.Either way, we weren't stuck for a charger, just disappointed that in my 5 years of driving EV's I've never seen a rapid charger only putting out 25KW. And yes I know if I'd waited longer than 10 minutes it would have started to increase.
Now I have Scan My Tesla installed via the OBDII adaptor I will be able to relate temps better to charge speed. What I have noticed is that in normal driving the pack stays around 20-21 C in cold weather so way off the 50 C targeted for maximum supercharging. It can take quite a long time to reach higher temps as I assume Tesla don't want to shock the cells with rapid temperature changes. There's normally a 1 C difference between "coolant" temp and pack temp which goes up to 4 C with pre-heating. Only reference I can give you is that the regen pips are gone by 22 C.
Also having just watched the latest Out of Spec Motoring vid on YT there's a lot of variability in charging speeds from several factors. Plus not all stalls at a site will necessarily be at the stated max in the info so try the highest numbered pair first I just in case they were the latest pair installed.