I feel I have to reply to this, as I’ve lived and trained, and ridden a bike, in London all my life, and actually live round the corner from Chalk Farm station, where you’ll be staying. It’s a fantastic place to live and the training options are endless. Quick summary:
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As people have said, take your bike if you can. You’ll only be able to rent rubbish bikes and they will be expensive. For a month, if you ride a lot, you’d probably be better off buying an entry-level road bike (about £500) and selling it when you leave, than renting.
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Swimming. There are lots of pools close to where you’ll be. All are drop-in, no monthly rates.
Parliament Hill Lido is best bet for summer training. 10 mins’ jog from Chalk Farm. 60m, unheated, aluminium-lined, beautiful when the sun shines, only £2 if you go when you should, between 7.30am and 9am (they throw you out at 9.30am and it re-opens at 10). No lockers in the mornings, don’t take valuables. If you go at other times it’s full of school kids who should be at school. It is set on the south end of Hampstead Heath, which is also, in my book, the best place in N London to run (if you like trails and hills). Men’s pond on the Heath is also great (I have heard…the women’s pond is shit for training). It has a 300m lap. You also might make friends there.
Standard indoor heated pools:
Swiss Cottage is the closest, but I don’t like it, lanes are v narrow and it is crowded.
Kentish Town is next-closest and better, although odd 30m lenghth.
Oasis in Holborn (Tottenham Court Road tube) which has 2 pools, one indoor (25m) and one out (27.5m) so if you go off-peak you can always swim
St Pancras Square, nr Kings Cross, few stops on the tube from Chalk Farm - brand new 25m but go off-peak and check the website for schools
Ironmonger Row, nr Old St station tube, again few stops on the tube - 30m, nice and clean, usually emptier than most
Marshall St, nr Oxford Circus, if you’re in the centre of town - 30m - tends to have a fast crowd in the fast lane, which is motivating
Someone mentioned London Fields. This is a 50m heated outdoor pool and is great. It’s a bit of a pain to get to,though, unless you ride there (40 minute ride through traffic).
These days nearly all public pools are run by Better Leisure (and I think every pool mentioned above is). Each of them has a good website - just google the name of the pool and you’ll see a timetable on the homepage. They all cost about £5 a visit.
Someone mentioned the Serpentine Lido in Hydo Park (Lancaster Gate tube is the closest). This is good open water simulation, because it is dark, full of weeds and other unmentionables. 100m length.
If you want drop-in coached swimming sessions good for triathletes (i.e. free style only), I go to a good one at Holloway Pool on Wednesdays 8.30 to 9pm (the pool is not open to the public the rest of the time).
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London is great to ride in. There are cyclists everywhere. Whoever said you’ll be riding indoors is nuts.
You are minutes away from Regents Park. Outer Circle Road, which runs round it, is N London’s unofficial bike race track. It’s a 3-mile lap, with some red lights and stops. There are always other cyclists to race, or follow, or keep you awake, so I don’t think it’s boring, even though it’s a hamster wheel.
You can do hill repeats on Swain’s Lane - runs up the side of Highgate cementary. Try it, and suffer (you need to descend down Highgate West Hill, as Swain’s is one-way). Very very steep, do 10 and you’re toast.
Richmond Park is great on weekdays. 7-mile lap, and it is lovely, and hard (hills whichever you go). Ride out (40 mins ride, but quite trafficy), or take the overground from Gospel Oak to Richmond (off-peak only, see below).
If you want open roads, take a train out from St Pancras (10 mins from Camden) to Harpenden or St Albans, 20 minutes journey and you are great riding country.
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Running. You have Regents Park, Primrose Hill and Hampstead Heath on your doorstep for running. Regents Park is flat, manicured, big. Hampstead Heath is much more wild, many trails, and has no flat at all. Even bigger. You’ll get lost there - take a phone with maps! Check out the Hampstead Heath Parkrun (parkrun.org) every Saturday at 9am - hilly 5km, free, good vibe, get a time. You don’t need to do much pavement pounding unless you want to.
You can run along the canal into town, and beyond (the canal in Camden goes all the way East to Stratford and the Olympic park). It’s a nice run - but it is concrete.
Talking of the Olympic park, there is a great LCM pool there (obviously!) and a fairly technical track for road bikes if you want some car-free miles (costs a few quid). Easy to get to from where you are: overground Gospel Oak to Stratford, or St Pancas to Stratford, quicker but pricier. You can take your bike on the trains, off-peak (i.e. between about 10am and 4pm, and after about 6.30 or 7pm), but never on the tube. (Ok, there are sections that allow bikes, but not in the centre of town where you are).
I think those are the basics. Happy to answer questions about the area, or what I’ve said above - PM me either now or when you arrive. Happy to go for a spin round Regents Park and talk it through!