It is a simple job, but the trick is generally to be gentle !
Because the space is tight I use a medium screwdriver that I bent about 3mm of the end at right angles - which I use to gently lever under each end between the chip and its socket a little bit until it moves.
If you tug them you can easily bend the pins, which are difficult to get in line again so they fit back in easily - the pins have to be exactly in line.
The more exacting bit is getting a chip in if its legs are not lined up exactly with the socket - and to help this I put the chip on its side on the metal lid and gently spring them closer together a little bit at a time.
I've had mucho practice at this, as I've been an electronics engineer for 25 years
And if it all goes pear-shaped - I have an open offer to anyone local that I'll either come round and help, or you send me the prom and if its still got all its legs I'll blow another one or straighten the legs properly.