Everything in "TV" is possible. You can watch British TV by using the built in browser on any MD. For cable, if you are allowed to decode using your own hardware then you can use some DVB-C cards with a CAM/CI unit and this will take the streams directly into LinuxMCE with no loss in quality. If not, then you will need to use capture cards and some mechanism to control the cable box (USBUIRT, serial, Ethernet, etc).
For radio - there is a mechanism for setting up Internet radio stations as scenarios.... its not very scalable but does work well. I think someone has put some thought into this on the Shoutcast side, but that doesn't really apply to normal Internet radio. FM/DAB - not sure what solutions exist to take these into the system digitally... I'm sure there is something, but worst case you can at least use LinuxMCE to control your existing receivers, change inputs and volume levels.
For HA the first two points are achievable using several different technologies... perhaps ZWave or KNX/EIB are the way to go for you. Don't know about controlling ovens and cookers... I suspect that isn't wise anyway, could be safety issues.
Telecom - all of that is pretty much standard in LinuxMCE, you just have to configure the Asterisk side of things, and that integrates with LinuxMCE. You access the voice mail through the LinuxMCE interface
For the security - you would need to add an SMS extension of some kind to Asterisk. After that there is no reason why you can't trigger any function you like (including lock/unlock doors) using a messagesend command to send the appropriate DCE message. There are some limited options for doing video cameras, not sure what you want to achieve with that, but certainly you can at least use Motion to trigger security events with video cameras. For the alarm system itself, there are several different alarm systems supported, or you can use Zwave/KNX/EIB HA devices and let LinuxMCE be the alarm system.
Yes, LinuxMCE can call mobile phones, etc on its security triggers and allow you to view security camera stills, speak to the house, etc.
Eee boxes and Eee Tops are box supported as MDs. Don't go down the ZigBy path, it isn't currently supported and discussions in the past have settled on the position that the return on investment in supporting it isn't worth it.
BTW: you are biting off far more than you can chew with this list of features. I would strongly recommend that you start with a small LMCE installation, and then slowly add the features as you become familiar with the system.