The vast Tube network extends far out across London's suburbs including Heathrow airport (Piccadilly Line). It's by far the cheapest way for you and your stags to get in to and around central London, with a single cash fare to central London (Zone 1) costing £6.00 and a one-day Travelcard for unlimited travel across Zones 1-6 costing £18.10 (Anytime) or £12.70 (off-peak).
Trains:Trains from Edinburgh to London hover around £100-£150 for a round-trip, creeping down the further south of the border your starting point is.
Taxi from airport to centre:Taxis from Heathrow, Gatwick, Stansted and Luton airports average at £50, £100, £80 and £75, respectively, book in advance to get cheaper fares. There are much cheaper options, like express trains (£10-£25), buses (£10-£15) and the Tube.
That said, if your stag group are travelling into Heathrow or Gatwick there are regular taxi transfers taking approximately an hour to each and costing approximately £50 to £80 per car. Better still is the Heathrow Express running straight to London Paddington and taking only 20 minutes, and the Gatwick Express arriving at Victoria. Cost-wise you're looking at £20 to £30 per journey and therefore if there's a group of you it's easy to run up a quick price comparison to see what's more cost-effective; for a group of 4 at that price, you'll likely end up in a cab.
If you're lucky enough to be travelling to City airport, which is based exactly where you'd expect; London City (docklands to be precise), by landing here could see you checking in to your central London hotel within an hour of disembarking the plane. City airport is well served by the DLR (part of London's tube network) and the taxi rank outside will have you rapidly relocated into town in a matter of minutes.
Stansted and Luton airports offer the choice of train, bus or taxi.
Other Flights in the UK:There are frequent direct flights from most major cities in the UK (Edinburgh Airport, Glasgow Airport, Manchester Airport, Belfast International) landing at any of the main London airports (Heathrow, Luton, Stansted, City and Gatwick). Flights usually take no more than an hour.