That's sorta the point of a uniform.
And not really. French uniforms were even more similair between regiments, as they had long abolished regimental colours.
But is there a british regiment that doesn't have the red-white-black combination?
Yes, that would be the every single British regiment aside from the 50th, 58th, 64th, 70th and 89th Regiments of Foot.
British regiment's had different colours on their uniforms called Facings. This was the collars, cuffs and shoulder epaulettes of their uniforms and would correspond to funnily enough the colour of their Regimental Colours.
For example the 1st, 2nd, 3rd Foot Guards and 1st, 2nd, 4th, 7th, 8th, 18th, 21st, 23rd and 42nd Regiments of Foot had blue facings whilst the 13th, 15th, 16th, 28th, 29th, 34th, 37th, 38th, 44th, 75th, 77th, 80th, 85th, 86th, 88th and 93rd Regiments of Foot had Yellow facings to their uniform.