Yes, a cancellation is 2 hours (so long as it is not extraordinary circumstances) hence you should be OK there.
I doubt you will have much problem with a Uber fare, particularly since you are Gold and there are mitigating circumstances. The worst outcome would be just £50 for the Uber, plus the 250€. Clearly if you accommodate another passenger or two on the taxi then you would be in a stronger position. Make sure you document this (e.g. screenshots and so on). Even at that hour the tube is likely to be the faster option depending on the final arrival point.
Here is the dispatch information which is a bit curious:
DOBA7338/22JUL
* OPERATIONAL FLIGHT INFO * BA7338 0 SU 22JUL18
CITY INFO HOUR (LOCAL)
PRG ESTIMATED TIME OF DEPARTURE 1930
DELAY ZY
FLIGHT CANCELLED 0829
CCRN
ESTIMATED TIME OF ARRIVAL 2022 LCY
*1A PLANNED FLIGHT INFO* BA7338 0 SU 22JUL18
APT ARR DY DEP DY CLASS/MEAL EQP GRND EFT TTL
PRG
1800 SU JCDRI/M YB/S E70 2:00
HKMLVNOQSG/S
LCY 1900 SU 2:00
COMMENTS-
1.PRG LCY - AIRCRAFT OWNER BA CITYFLYER
2.PRG LCY - OPERATED BY BA CITYFLYER
3.PRG LCY - DEPARTS TERMINAL 1
4.PRG LCY - 9/ NON-SMOKING
5.PRG LCY - ET/ ELECTRONIC TKT CANDIDATE
6.PRG LCY - CO2/PAX* 147.08 KG ECO, 147.08 KG PRE
(*):SOURCE:ICAO CARBON EMISSIONS CALCULATOR
CONFIGURATION-
E70 C 16 M 60
The cancellation code is CCR and N stands for no compensation payable. I thought CCR is crew shortage which almost always leads to EC261, but hopefully someone can throw some more light on the reasons for the cancellation of BA7338. It doesn't stop you claiming the taxi and right to care expenses. I would urge you to find out what the ground crew at PRG know.
I hope you feel better soon. I had an odd experience on Friday when I felt unwell on boarding but fine by arrival, which in shorthaul is not to be expected!