There are bargain tickets available for the Heathrow Express (LHR airport to Paddington station) if you buy your tickets early - from 90 days to 6 months in advance. A one-way ticket is £6.99; return ("round trip" for us Yanks) is only £11.90. This is a real deal; it's cheaper even than the Heathrow Connect, and is much faster. Here's a link:
https://www.heathrowexpress.com/tickets-deals/discounts-deals
There are more details in a thread on the London page in this forum entitled "UK: getting my 3 basic ground transportation cards."