Getting from Heathrow to Balham with heavy luggage (mid morning sunday 14 December)

Step-Free Access Research

Just for kicks I used tfl's journey planner, and selected the option "no stairs or escalators". I could not use your planned date (10th December 2014) because you can only plan ahead one month (so it'll have to wait for tomorrow. Here is the link to the suggested route. Below is a screenshot:

The "no stairs or escalators" is a fancy way for TFL to say "step-free access", i.e. with elevators from street to platform. It might be what you are looking for since you mention heavy luggage.

On a different note you could ask for help with your luggage to any of your fellow travellers. I bet you could easily find someone willing to do so. ;)

Tube It for £5.10

Then I went to the single fare finder page and searched for the cost from Heathrow T1-3 to Balham. Off-peak is £5.10. Does this fit your budget?

Plan Ahead for Works

As a general rule of thumb I advise you to check TFL's planned works calendar page. If you are landing at Heathrow you might want to be informed as early as possible that the Piccadilly Line is closed/broken. And since most of the works take place on weekends, tourists tend to be the ones more affected by the disruptions.

Tube, every time. (see below)

Balham has escalators. http://www.tfl.gov.uk/tube/stop/940GZZLUBLM/balham-underground-station?lineId=northern

Victoria has escalators: http://www.tfl.gov.uk/tube/stop/940GZZLUVIC/victoria-underground-station/

All deep-level tube stations have lifts or escalators, except in cases of mechanical failure which are super-rare. Sometimes there are short staircases to get from platform to escalator (but not in the case of Balham IIRC).

'Step-free' access isn't really relevant to you, so long as you can carry the case up a short flight of stairs. 'Step-free' access is for people in wheelchairs for whom (particularly in the case of electric wheelchairs) even 1 step can be an insurmountable obstacle. It doesn't sound like you fall into that category.

If you absolutely can't carry the case at all then you're better off getting a taxi. Heathrow has dozens.

BUT - checking your route - to go by tube from heathrow to balham with heavy luggage, you don't want to be changing at Victoria (which would take you onto the 'big train' - no escalators at Balham, I think no lift either) or changing at Green Park (which would mean you'd have to change again at stockwell - I'm assuming that with heavy luggage you'd rather spend longer on the tube train but have fewer changes).

Tube map is here: https://www.tfl.gov.uk/cdn/static/cms/documents/standard-tube-map.pdf

What you want is Picadilly Line then Northern Line.

So change at Leicester Square if you can manage a few short flights of steps, Kings Cross St Pancras if you can't.

I've just checked and Balham does have lifts from the 'big train':

http://www.nationalrail.co.uk/stations/BAL/details.html

Step-free access to platforms 1 and 2 from street via lift.

so
- Heathrow -> tube to Victoria -> big train to Balham
is a realistic option, and might be the quickest way to do it, although the change at Victoria won't necessarily be entirely stairless.

This is probably your best option. Trains from Victoria to Balham look pretty frequent even on a Sunday afternoon.