Thyme or Speedwell Groundcover

I live in Toronto and have small city backyard. I also have two female dogs and a 6-1/2 year-old son. About one-third of our garden is a vegetable plot. The rest is an on-going disaster. I am interested in trying to see whether I could cover the remaining two-thirds with groundcover, and in particular am considering woolly or creeping thyme, and/or creeping speedwell. Do you have any thoughts on whether this groundcover idea has some prospect of working or whether it is an idea doomed to fail? If it is worth trying, do you have any views as to whether thyme, speedwell or a combination would be preferable?

You do not mention whether the area in question is shaded, full sun, or something in between. Because part of the garden is a vegetable garden, I will assume that the exposure is full sun or mostly sun through the day. If not, then thyme will not thrive as well.
Much depends on exactly what you are trying to achieve and on what effort and money you plan to invest. Dealing with the money side first, both woolly and creeping thyme cannot be grown from seeds, so you would need to buy plug trays. One plug tray has 120 cells or plantlets which is enough to cover as much as 30 square feet when spaced six inches apart. At a six inch spacing, it will take about a year to fill in. At a closer spacing, say four inches, the plants will fill in quicker but one plug tray will not cover as much area (only 13 square feet), consequently the cost is more than twice as high.
Woolly and creeping thymes are very low. They provide a very nice, neat, and reasonably even surface. They do require more care than the next alternative described below, because they do not compete as well against weeds – so you need to pull weeds and grasses out from time to time.
Wild thyme is the one spreading thyme that can be grown from seeds. You could sow the seeds directly over the area, or you can purchase plug trays. The seeds are very fine and seeding them and getting them to germinate and develop is trickier than, say, lawn grasses. However, once established wild thyme is a vigorous and hardy plant that will quickly fill in and give you a thick cover. It is higher, reaching six to eight inches compared to the two inches of creeping and woolly thyme. But a big advantage is that wild thyme competes the best of all the spreading thymes against weeds and grasses, literally choking them out over time.
Whether you plant the low thymes or wild thyme, you need to prepare the area very well. It must be free of weeds, otherwise you will be battling weeds incessantly.

And finally, is the Veronica officinalis listed in your catalogue a creeping speedwell?

Yes, common speedwell (V. officinalis) is a creeper. It does well in full sun and partial shade. It can be started from seeds, but we do not recommend direct sowing in the ground; it is better – because speedwell is slow and sporadic to germinate – to sow in flats or plug trays first and then transfer to the intended area later in late summer or early fall.

P.S. If you say it is worth trying, I plan to buy the plants from you (if you have them). How soon would I have to place my order to ensure being able to receive enough plants for the end of May?

Generally, because demand for plug trays is so high in spring, you need to order at least eight weeks in advance. Some varieties may be in better supply at times, but it is best to call for an update on what is available and when. Seeds are available anytime.
Note that there is no reason why you cannot start your project later, say in June or July or even later. All thymes and speedwell are hardy and adapt well to summer planting if they get plenty of water during dry spells.