Thumping into gear

MODERN engines are designed to be driven away from a cold start with no warming up, so doing that doesn't damage the engine. Your problem is probably the carburettor or the supply of warm air. Warm air is normally piped into the carburettor in the first few minutes of driving, and this is often dumped when the engine is worked on later in life. It could also be wear in the carburettor or simply an incorrect adjustment of the carburettor or choke. Have the carburettor settings checked, and check it for wear.