I am a big fan of swedes/rutabagas.
When I can't get them I use turnips, beets, daikon radishes.
Carrots/parsnips add some interesting flavor as well.
A couple other vegetables that are crunchy and nice-flavored but more exotic are celery root or jicama.
cooking.stackexchange.com/users/4428/J. Win.
turnips will lend the necessary flavor but can be added WITH: Cauliflower or slightly cooked potatoes such as Petite Potatoes (grade C which you can find at a farmer's market but pack some taste...yum), Yukon Gold, New Potatoes, Red Potatoes or Fingerling Potatoes.
I know some people who have substituted rutabagas in recipes with the stronger portions of a green cabbage
(do NOT use the core).
I came from a poor community and you learn what you can use and not.
Things that grow in your garden like: the aforementioned vegetables and brussels sprouts, cooked parsnips, carrots, corn, peas, radishes, zucchini, cooked butternut squash, toasted or grilled eggplant or cheaper items at the store like jicama, cabbage and celery.
Don't forget that some raw slivered (not already roasted) almonds, sunflower seeds or cashews will give you some more crunch too.
I know that this is an old post but some people will find this in a search for the same issues like I found in my Google search...
so Good Luck to all :)
cooking.stackexchange.com/users/37020/PeggySue2u2
Anything crunchy should work fine.
Radish, daikon (yes I know it's a kind of radish), carrot, fennel...
cooking.stackexchange.com/users/None/
water chestnuts might do the trick for crunch.
cooking.stackexchange.com/users/15/mfg
Kohlrabi is a brilliant idea and it will not at all seem much different to the swede in a Branston Pickle type mix which has a wide variety of vegetables in it anyway.
Turnip will do at a pinch but you may need to increase the amount of sugar as swedes are generally sweeter.
Make sure the chunks are very small.
Here is an original recipe which comes out a little dryer than the commercial Cross & Blackwell brand.
Some people add tomatoes and more apple in the mix.
If you can't find fresh gherkins straight off the vine, buy a cheap commercial jar of them - dice and throw in for good measure.
At the end of the day it's finely chopped dates you should want rather than raisins, and flavour comes from the dark brown sugar (Muscovado is best), All spice, standard dark brown malt vinegar, and of course cayenne pepper.
Make sure you keep the mix moving, with sufficient liquid.
Do not allow the pan to burn at the bottom because the taste will quickly change.
If you do catch the bottom of the pan, do not scrape it back into the mix.
Decant the content immediately into another pan and leave the burned bottom behind.
Also, once the product is in the can, and sealed, do not open and use it for at least a month, to allow the blend to mature (all the flavours will even out naturally over time).
9 ounces carrots, peeled, cut into small chunks 1 medium swede, peeled, cut into small chunks 4-5 garlic cloves, peeled, finely chopped 5 ounces dates, finely chopped 1 small cauliflower, finely chopped 2 onions, peeled and finely chopped 2 medium apples, unpeeled, finely chopped 2 medium courgettes, unpeeled, finely chopped 15-20 small gherkins, finely chopped 10 ounces dark brown sugar 1 teaspoon salt 4 tablespoons lemon juice 3/4 pint malt vinegar 2 teaspoons mustard seeds 2 teaspoons ground allspice
1 teaspoon cayenne pepper
cooking.stackexchange.com/users/61672/Stephen Nicholson
Could you use a turnip (or several)?
My understanding is that this is the "closest" vegetable to the rutabaga.
cooking.stackexchange.com/users/4980/Aaron
I want to make a British-style pickle relish at home. I think the best recipes involve swede (rutabaga in the US), because it gives texture and flavour, but isn't overpowering. Unforunately, I can't get swedes where I live.  What would make a good substitute?
What can I use instead of swede (rutabaga) when making pickle relish
cooking.stackexchange.com
<substitutions><vegetables><chutney>
cooking.stackexchange.com/questions/12813
cooking.stackexchange.com/users/611/Carmi
Beets, daikon radishes, stronger portions of a cabbage, carrots or turnip mixed with cauliflower or slightly cooked potatoes will add the necessary flavour, or you could try other vegetables from the garden like brussels sprouts, corn, peas, radishes, zucchini, cooked butternut squash, toasted or grilled eggplant. Try using turnips, daikon, beets, carrots, parsnips, water chestnuts and kohlrabi. 
Try using turnips, daikon, beets, carrots, parsnips, water chestnuts and kohlrabi. You might need to add some sugar to replace the sweetness of the swede.