I'm sure those businessmen are happy paying £30+ for a 15 minute journey (if those businessmen are not on expenses, of course)
Very much depends on the business. Looking now, a Heathrow Express ticket is £25 one way, a non-Heathrow Express ticket is £10.50, a premium of £14.50. The average journey times are 21 minutes (4tph, average wait 7.5 minutes) and 34 minutes (2tph, average wait 15 minutes) - so travelling by Heathrow Express saves 20.5 minutes on average. That's without considering the fact that the services go to different terminals, and the associated transfer time between T4 and T5 if required.
If our businessman's time is worth more than £42.44 an hour, then Heathrow Express starts looking sensible - provided that travel time is bookable to the customer. It isn't always, depending on the contract, but it may well be. When I did a bit of project management work seven years ago, we valued skilled tradesmen's time at £55 an hour, so Heathrow Express isn't ludicrous.
That said, I'd imagine that most companies will dictate that travellers use the most cost-effective means to make their journey.
That is not true. While there is no justification in companies paying for first class for staff to 'have a jolly' if it is the difference between a place work can be done and getting no productive time out of a travelling employee they will be happy to authorise it.
Increasingly, it's as easy to work in First Class as in Standard. Smartphones have seen to that - they're perfectly suited to emails, and can be used to tether a laptop if free WiFi isn't available.
And that assumes that work can be done whilst travelling. That isn't always possible - the work may require specialised equipment or software that isn't available on a train, or there may be restrictions on how company (or customer) information is handled that make it inappropriate to do so.
A good many will not authorise First Class at all these days.
I did once travel First Class on Virgin Trains (as a lowly graduate, no less). It was only possible because company policy dictated that the cheapest ticket should be bought - and for the times and dates I needed to travel, there were First Class Advances left but no Standard Advances. Justifying it was surprisingly easy (screenshot of the journey planner showing that no cheaper option was available) considering that company policy generally banned First Class rail travel, and only allowed business class air travel when flying longer than 4 hours
and expecting to work immediately on arrival.
A colleague who worked for the same company got a return First Class sleeper journey expensed, because he couldn't fly on medical grounds, had to be at an early meeting, and company information security rules meant that he couldn't have a shared sleeper berth if he had his laptop with him. The company rules on pay for travel time then meant that he was receiving time-and-a-half whilst sleeping at 80mph. On two consecutive nights. Unsurprisingly, he wasn't sent to such meetings in the future, but on that occasion there was no alternative.
If the company is sensible (and in the case of the above company, I'd hesitate to use the term) there may well be a general ban, but there'll be a process to make exceptions where appropriate.