For any commercial installation, use commercial equipment. Trust me, it will save you a lot of heartache in the long run! Speakers in particular feature much better protection from abuse and are much easier to safely mount.
HDMI will run at 1080p up to about 30M with active cables, so as long as your cable runs are less than this, you should be able to run from a Sky, Freesat or Freeview box. I would suggest you also have an input for a laptop or similar - which will give you good flexibility in terms of streaming from other sources.
Denon make a great AV receiver designed for installations. Its has Bluetooth for streaming audio and can be rack mounted - which means you can put it into a cabinet to control access, make it harder to steal and also tidy the cables. -
DN-700AV Professional 7.1 AV Receiver
Link this to a good quality HDMI distribution splitter and you should have an excellent, stable picture with the ability to switch to various sources.
Speaker wise, these:
LT8P are a good, solid speaker with good mounting hardware and they sound pretty good as well! You could add a sub if you wanted to, but unless you are really pushing the system, I don't think you will need it.
Screens - again I would go for commercial screens - possibly with toughened glass, but you need to weigh up how boisterous things get!! Commercial screens are generally a bit brighter and are rated for longer operational periods - usually specified as 24/7, 8/5 etc. Make sure you get solid mounts and they are installed correctly. I've seen too many bodged installations in pubs and clubs and the results when a screen does decide to come down!
£800 is not enough budget for the whole system. I think the amp will be this on its own and the speakers are about £200 each. If you need to add in the HDMI splitter and the screens - price depends upon size, Freeview / Sat box, any mains work that needs doing, cabling etc and I think £3K - £4K is a more realistic budget.
Could you do it for £800? Possibly, but the result will probably have quite a short lifespan and you will be replacing speakers and probably other parts quite quickly. You might also get hums and buzzes from trying to get audio out of the TVs and issues setting up sensible sound levels and changing channels without juggling remotes.