Green’s Dictionary of Slang

1. to pursue (a matter or person) vigorously with a specific intent, esp. after an earlier unsatisfactory response.

1974		G.F. Newman You Flash Bastard 100: ‘Anything on those prints?’ ‘Not a word yet, guv,’ the huge DS replied. ‘Well, chase them up.’.
1981		A. Payne ‘You Need Hands’ Minder [TV script] 34: Well, I don’t like it, Terry, being chased up in the middle of a big job like this.

2. to make efforts to find or obtain quickly.

1977

G.F. Newman Villain’s Tale 87: He was satisfied that the investigation was progressing. He would get all the details from the report and wouldn’t need to chase it up personally.