The National Rail website is coming up with these times and prices (all for 'Advance' tickets):
Taking the RailAir Coach to Reading and then a train to Taunton:
Depart 1725, arrive 1948, £49.50
Depart 1825, arrive 2054, £35.50
Taking a train Paddington and then one to Taunton:
Depart 1757, arrive 2046, £44.50
Given the possibility of a delayed flight, and the lowest fare, I'd be tempted to allow time for a coffee at Heathrow and then take the 1825 coach+train route. However, if you're going that way anyway, you can still do it cheaper, as follows:
If you book the RailAir coach at least two months before travel, the fare from Heathrow to Reading is just £15.00. You need to specify your date of travel when booking but not the specific service which you'll be travelling on. There are departures from Terminal 5 to Reading as follows:
1705 (arr 1754)
1725 (arr 1814)
1745 (arr 1834)
1805 (arr 1854)
Any of those services gets you to Reading in time for the 1903 train to Taunton, which is the first of several offered at just £13.00. (Arrives in Taunton at 2046).
So my recommendation is to purchase a coach ticket to Reading for £15.00 and an Advance ticket on the 1903 service from Reading to Taunton for £13.00, making £28.00 in total. If your plane is on time (and you take the first coach to Reading) you'll have time for a coffee and a sandwich at Reading Station (or, better still, a drink at The Three Guineas pub, which is built into the station!). If your flight is slightly delayed you'll still arrive at Reading in time to catch your train.
Book your coach ticket here:
http://www.railair.com/index.php
and your rail ticket here:
http://www.nationalrail.co.uk/
Whatever you choose, don't leave it until the last minute as the off-peak 'walk up' fare is £51.00 via Reading or £65.40 via Paddington!