I believe it's only pedestrians that are banned from the tunnel - they can of course travel free through it on local buses.
In which case they are not pedestrians. I only got that same map - dated 2008, but also talking of 400+ cycle parking spaces around the site, including the central area.
I doubt they'd be able to just ban all cyclists from coming in - there's be a forum thread on that somewhere, we're an angry bunch when we get shut out!
Also, there's the Equality Act, under which someone could argue that if they (through disability, health condition etc) were unable to drive a car, they had a basic right to the next most flexible means of getting around, which is the bike.
The "left turn" issue has been discussed before, either on this thread or others (Fantasy Island & so on). There are two key problems:
1) The entire airport complex is in tunnels and not designed for diesel stock. Therefore the left turn is a non-starter until the line is electrified - now under way.
2) Long distance trains have traditionally been 8-10 coaches long. The HX is 5-coach, does anyone know the platform length at each of the 3 stations?
My understanding is that the current Vomiter, sorry, Voyager fleet will be lengthened by adding on a coach with pantograph, so this would take the super-voyagers to 6. I think most of the HST replacements are also 7+.
So a regional service would be more likely in the interim, perhaps terminating at Oxford?
After that, is there not supposed to be a direct link from the GW line into T5, enabling a continuation from there in to Paddington? Announced a few months ago, not seen any details. This would presumably allow for full length (10-11 coaches?) trains.
Beyond this is the prospect of an HS2 link, but only in Phase 2. Frankly, even though I admit to being sceptical about the whole HS2 deal (who wouldn't be with another £10bn on last week), I really can't see the link into LHR happening.
For these trains, you are talking double length, even double height. The tunnelling requirements would be huge, so the talk is just of serving T5. Then you have the issue of track paths - the most important destination from LHR is central London, which is served by 4x5 coaches hourly. The West Midlands are what - 2-5% of volume? So you are already spread very thin. Brum's trumpeting of HS2 blows up in their face if there's a direct link sucking more people into LHR, but would these trains be viable? Even if there is a business case - based on £4bn for an hourly service, v £20bn+ for Phase 1, but an eventual capacity of 18 trains / hour, what would Beardie / Delta / Star / basically all the other airlines say about the unfair advantage going to BA?
Maybe they wouldn't be bothered because that's balanced out by T5 already being further from London than the other terminals. Maybe they wouldn't be bothered because they just wouldn't see a threat.
Either way, none of the "obvious" rail answers are as obvious as they seem when you deal with UK rail economics.