Will the bricks on my extension ever match the rest of the house?

Honestly, an exact match is rare, and any bricklayer who promises one straight off is overselling it. Bricks are made from natural clay, so colour and texture shift from batch to batch, and the bricks already on your house have weathered for years. New ones next to them will always look a bit fresher at first.
How close you get depends mostly on the age of your home. Houses built after about 1940 use metric bricks, which are much easier to source a match for. Anything older tends to be imperial size — a touch larger — and those are trickier, sometimes needing a specialist reclaimed-brick merchant.
Two things people forget. Mortar matters as much as the brick: get the colour or the joint style wrong and even a spot-on brick looks off, so match the mortar too. And ask your bricklayer to build a small sample panel, a square metre or so, then let it dry before you sign off the whole elevation — wet mortar and dry mortar are different colours.
If no good match exists, brick tinting is a well-established fix in the UK, where new brickwork is hand-coloured to blend with the old. Time helps as well: fresh brick usually weathers into its surroundings over two to five years, so a small difference on day one often fades on its own.
Take a brick (or clear daylight photos) to a couple of local merchants before work starts, and see what bricklaying costs while you're planning. If you'd like a few quotes from vetted local bricklayers who've handled matching work, we can match you with tradespeople through BestBuilders.