There aren't any where near as many budget chain motel/hotels as there are in the US. The only one I can think of is Travelodge, and is among the cheapest you are likely to find in the UK. The ones that I know of are fairly new, and their pitch seems to be the cost-conscious business traveller, and so I guess would have modem links in the rooms. There is one close to Heathrow, and I think that the room charge is £70 a night (incl tax - NB UK room rates always include tax). Others are cheaper, like the one in outer SW London in Morden (probably too far out to be convenient for central London), which was £60 a night the last time I looked.
There are also HolidayInn Express hotels, which are also reasonably priced (for the UK

). There is one in SW London at Wandsworth, just 15 minutes out of Waterloo station by train and right opposite one of the three (or is it four now?) Outback's in the UK. The room rate there is about £90 a night. There is another another a bit further south in Colliers Wood, very close to the Northern (black) tube station, which has roughly the same room rates.
Alternatively there are a miriad of privately owned hotels, many of them historic/older, dotted around the UK, with welcoming owners/staff and often very reasonably priced, but you'd almost certainly have to forgo your modem link.
You can be sure that ALL hotels in the UK will offer non-smoking rooms.
Do you know where in the UK you will be staying?