There is a Mr Timpson's in the Arena next to Sainsbury's. I've got them to fit watch batteries for me a few times.
If you're on Seven Sisters side, the hardware etc shop over 7Sis station (High Rd side) will sort it for you.
My go - to place is the jewellers right by the traffic lights opposite Bruce Grove station, but nearer to you would be upstairs in Wood Green shopping centre, one of the stalls sells watches and will change batteries.
Don't bother paying someone to fit a new watch battery. It is easy, using a pen-knife/Swiss army knife, to ease off the back of a watch, and then the battery is staring you in the face. You might need a magnifying glass to read the number on the back of the button battery. Then go to Lido on Green Lanes, who keep watch batteries in stock, and buy one for a half or a third of what you'd pay someone to fit it for you. Slide it into place, use your thumbs to press the back on, and it is all done.
I go to Erbilla jewellers either at Bruce Grove or Wood Green. I have done the replacement myself with shop bought batteries but have found they don't last as long as the ones the jewellers stock & I find it hard to get the back on again properly. I must try this Lido place though that Christopher Fowler mentions.
The best quality button batteries for watches are Renata, made in Switzerland. Lido has them.
I go to Ozcan Jewellers on the corner of Umfreville Road and Green Lanes.