Replacing Text In A Column For All Rows In The Table?

I have a column like this:

Table: User
name, field1
Jack 1000|1001|1003
Berg 2000|1001|2004
Paul 1001|1000
Jane 1001

Now, I would like to replace all "1001" with nothing, and also remove the "|"-separator behind 1001 if it exists, basically removing both "1001" and "1001|", so the resulting table looks like this:

name, field1
Jack 1000|1003
Berg 2000|2004
Paul 1000
Jane

My tries, been plenty, but here are some:

Code:
UPDATE UserTable
SET field1 = REPLACE(field1, '1001', '')
UPDATE UserTable
SET field1 = REPLACE(field1, '1001|', '')

But the above queries replaces field1 only if the whole field matches '1001' or '1001|'...

The above queries do work, just like I want them to. I just happened to write them in this order on this post... did not do a copy of the actual query.