Although at first glance Crossrail 1 (the Elizabeth line) won't have any impact on Virgin Trains East Coast (VTEC) when it opens in 2018, it does have the potential to make air travel more attractive as it will make Heathrow more accessible from Central and East London. It will also make towns in the Thames Valley such as Reading and Maidenhead more accessible from London.
Currently getting to Heathrow from Central London can be a pain and although London City Airport is more convenient for those in Canary Wharf or further east it does not have the range of flights that Heathrow does. Crossrail will make it possible to get a direct train to Heathrow from Central London, Canary Wharf and points further east. Fares will be fully integrated with other TfL modes so Oyster and Contactless will work. As Crossrail is not a tube line the trains are more spacious so carrying luggage is less of a concern.
Currently there's 3 rail based options to Heathrow. The tube is by far the cheapest but during busy periods you're unlikely to get a seat and travelling with luggage is bothersome. The most expensive option is the Heathrow Express that runs non-stop from Paddington in 15 minutes, every 15 minutes. These trains are comfortable and have lots of luggage space but unless you live close to Paddington you still have to get there which can involve a crowded tube and connecting from the tube to the train in Paddington with lugage isn't an ideal experience. The Heathrow Connect is a mid price option, the train takes about 10 minutes longer than the Heathrow Express and runs only every 30 minutes, as it also starts at Paddington it has the same accessibility drawbacks as the Heathrow Express and also the addition of the trains being more crowded as it's also used by commuters on the intermediate stops.
Even if you don't live near a Crossrail station it should still mean fewer changes will be required to get to Heathrow and even though Crossrail is not a tube line changing between Crossrail and the tube will be similar to changing between tube lines as there's no need to go through barriers when changing lines. Once Crossrail opens it will replace the Heathrow Connect service, but the Heathrow Express will continue to run.
So on one side Crossrail will help increase the attractiveness of flying as it will make getting to Heathrow simpler and provide a much needed capacity boost. Domestic flights from Heathrow currently serve Newcastle, Leeds, Manchester, Edinburgh, Glasgow, Aberdeen and Inverness (as well as Belfast but that's not directly competing with the rail network) so making flying more attractive could have an impact on rail usage. There's three other new developments which may also have an impact: Flybe are starting flights from Heathrow to Aberdeen and Edinburgh providing competition with British Airways on these routes, British Airways are launching their Club Europe business class product on domestic flights from 1st April 2017 and also British Airways in the next few months are opening a new 'First Wing' fast track with direct lounge access for first class passengers and gold card members, retaining the existing fast track lanes for other qualifying customers which should reduce the wait at security.
Virgin Trains East Coast has a strong product (particularly in first class), excellent crews (on the whole), a better catering selection and provide a generally relaxing experience throughout the whole journey. The thing that lets them down is the loyalty scheme. With East Coast Rewards there was a scheme that showed their regular customers that they were valued. British Airways has the Executive Club that gives particular benefits based on a tier you reach based on the amount of flying you do as well as earning points to spend on flights, hotels, etc. If someone earns Executive Club status through their work travel they can then use the benefits on their personal travel too, this has to be very attractive to business travellers when deciding whether to go by train or plane.
If VTEC keep up a decent product offering at a competitive price then the opening of Crossrail is likely to have minimal impact on VTEC but now is the time to try an win over flyers by offering a loyalty scheme that really encourages rail travel and have them using VTEC long before Crossrail opens.
There is also an opportunity for VTEC to target passengers in the Thames Valley when Crossrail opens. Currently if you live in Slough, Maidenhead or Reading the easiest option is to jump into a taxi at Heathrow and it's a relatively short drive along the M4. There's currently no rail line going west from Heathrow (although it is planned) so getting the train involves either going back to Hayes or Paddington and getting a westbound train from there. This is still preferable to having to get the tube across from King's Cross to Paddington but Crossrail will make it easier to connect to direct trains to the Thames Valley from Farringdon which is just one stop away from King's Cross.