The simplest way to get from LHR to Cambridge is to take a direct National Express coach (highway bus). The fare is £27. Alternatively you could take the Tube from LHR to King's Cross St Pancras for £4 and a train from King's Cross to Cambridge for £18.
To go from Cambridge to Bath Spa requires taking a train to London King's Cross, the Tube or a taxi to Paddington, and a train from there to Bath Spa. The walk-up train fare is £62. Booked well in advance on the National Express East Coast site, the fare can be as low as £13. The site accepts North American credit cards. The Bath Spa-London walk-up fare is £47. Booked well in advance online, the fare can be as low as £13.
The Tube fare from anywhere in central London to LHR is £4. Or you can take the Heathrow Connect train from Paddington to LHR for £6.90. Or you can take a National Express coach directly from Victoria Coach Station to LHR for £4.