We walk in the dark places no others will enter
We stand on the bridge and no one may pass

declare @server_id int 
declare @sql nvarchar(max)

set @server_id = -1
while exists (select top 1 1 from sys.servers where server_id > @server_id)
begin
	
	select top 1 @server_id = server_id, @sql = QUOTENAME(name) 
	from sys.servers 
	where server_id > @server_id 
	order by server_id
	
	set @sql = 	'select * 
	from 
	(
      select QUOTENAME(bs.server_name) + ''.'' + QUOTENAME(bs.database_name) DBName
      , bs.type BackupTypeCode --included since it looks like this field could be null, so not restricted to the lookup values
      , t.descrptn BackupType
      , COALESCE(Convert(nvarchar(20), backup_finish_date, 113),''Backup Not Taken'') FinishDate
      , COALESCE(Convert(nvarchar(20), backup_size, 101),''NA'') Size
      , COALESCE(Convert(nvarchar(20), compressed_backup_size, 101),''NA'') CompressedSize
      , COALESCE(Convert(nvarchar(20), backup_set_id, 101),''NA'') SetID
      , name BackupName
      , ROW_NUMBER() over (partition by bs.database_name, bs.type order by bs.backup_start_date desc) r
      from ' + @sql + '.msdb.dbo.backupset bs
      left outer join 
      (
            select ''D'' id, ''Database'' descrptn
            union select ''I'' ,''Differential database''
            union select ''L'' ,''Log''
            union select ''F'' ,''File or filegroup''
            union select ''G'' ,''Differential file''
            union select ''P'' ,''Partial''
            union select ''Q'' ,''Differential partial''
      ) t on bs.type = t.id 
	) x where r = 1
	order by DBName, BackupTypeCode
	'
	
	exec (@sql) --you could insert to a temp table / something like that to make use of the results later
	
end