Yes, American DS games work perfectly on a European (UK/EU) DSi. Standard Nintendo DS cartridges are region-free.
DS game cartridges (grey/black) lack region encoding. They run on any DS, DS Lite, DSi, DSi XL, 3DS, 2DS regardless of hardware region.
For 95% of the DS library (standard carts), region is irrelevant. Only the small DSi-exclusive library (≈40 titles) enforces region locking.